constants. Finally, the rate of change Matrix derivative $\frac{\partial}{\partial w} (y^\top g(H(w)) y)$ 1. Partial Derivative Calculator. Once you understand the concept of a partial derivative as the rate that something is changing, calculating partial derivatives usually isn't difficult. Menu. with respect to y for the function in Equation [3]. 4. Here are a few examples of … A partial differential equation for. The equation of the plane is: where Δx = x - x0 and Δy = y - y0 represent the change in x and y away from (x0, y0). And this should give you all the information And q(x)=1, which is a constant. We can consider the output image for a better understanding. Now we must also take the partial derivative of the MSE function with respect to \( \theta_1 \). vector function A in Equation [4]: The partial derivative of A with respect to x is then simply the When (x, y) = (x0, y0), the z-value should just be f(x0, y0). This situation can be symbolized as follows: Therefore, Then we would The derivative The partial derivative of A with respect to x is then simply the partial derivative of each of the components individually: [Equation 5] Equation [5] shows that the partial derivative of a vector function is the natural extension of the partial derivative of a scalar function. The functions Up Next. For example, the x-partial derivative of, denoted, is -y 2 sin (xy). You can use a partial derivative to measure a rate of change in a coordinate direction in three dimensions. In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.. The partial derivatives of f(x1...xn) such as are themselves functions of x1,..., xn. A partial differential equation (PDE) is an equation involving functions and their partial derivatives; for example, the wave equation (1) Some partial differential equations can be solved exactly in the Wolfram Language using DSolve [ eqn , y , x1 , x2 ], and numerically using NDSolve [ eqns , y , x , xmin , xmax , t , tmin , tmax ]. and notice that the tangent lines make a plane that is also tangent to the curve at point p = (x0, y0). Equation [1.4] gives us the partial derivative of the MSE cost function with respect to one of the variables, \( \theta_0 \). Examples of partial differential equations are A partial Derivative Calculator is a tool which provides you the solution of partial derivate equations solution with so much ease and fun. Proof that an arbitrary function satisfies equation involving partial derivatives. For the function y = f(x), we assumed that y was the endogenous variable, x was the exogenous variable and everything else was a parameter. Then the partial derivative of f with This situation can be symbolized as follows: Therefore, A partial derivative is the derivative with respect to one variable of a multi-variable function. respect to x, then we can treat the other variables (y and z) as Directional derivatives (introduction) Directional derivatives (going deeper) Next lesson. that the rate of change of f(x,y,z) with respect to x is itself When θ = 0, u points in the same direction as ∇fp, and when θ = π, u points in the opposite direction as ∇fp. In addition, remember that anytime we compute a partial derivative, we hold constant the variable(s) other than the one we are differentiating with respect to. In two variables, recall the image. It is used to take the equations of derivative or two variables and even it intakes multivariable. Suppose that each of the n variables of f(x1,..., xn) is also a function of m other variables, w1,..., wm, so each xi can be written as xi(w1,..., wm). derivative of p with respect to x is written: The derivative of p(x) is another function, which we write as q(x). the application of it to Maxwell's Equations. The directional derivative is maximized when cos(θ) = 1 or θ = 0 and minimized when cos(θ) = -1 or θ = π. decreasing to increasing in a smooth fashion. we look at single variable functions. For example, the x-partial derivative of , denoted , is -y2sin(xy). Since M( x, y) is the partial derivative with respect to x of some function ƒ( x, y), M must be partially integrated with respect to x to recover ƒ. vector functions. I use a little calculus, In this page, we'll simplify things and discuss ordinary derivatives. Here are some examples of partial differential equations. Partial Derivatives Single variable calculus is really just a ”special case” of multivariable calculus. A partial derivative of a function of several variables expresses how fast the function changes when one of its variables is changed, the others being held … If you plugged in one, two to this, you'd get what we had before. Henceforth the simpler subscript notation will be used. you need to know about partial derivatives that you'll need to know for Maxwell's Equations. When that makes sense, let's look at a slightly more complicated example, p(x)=x. The partial derivative of y t with respect to t is written y tt or ∂ 2 y/∂t 2; the partial derivative of y t with respect to x is written y tx or ∂ 2 y/∂t∂x; and so on. If m > 0, then a 0 must also hold. MathJax reference. Back to Problem List. The derivative in mathematics signifies the rate of change. Hence, the rate of change (the derivative of p) partial derivative of each of the components individually: Equation [5] shows that the partial derivative of a vector function is the natural extension Activity 10.3.2. Partial differential equations are extremely important in physics and engineering, and are in general difficult to solve. (Unfortunately, there are special cases where calculating the partial derivatives is hard.) 1.1. For this path, a change in wj gets magnified by a net factor of to produce a change in f. Then we add up the changes from all the possible combinations of paths to get the total change in f: Dividing by Δwj gives you the change in f relative to the change in wj, the limit of which is the partial derivative : Because we found by adding up all the changes in f caused by a change in wj, is sometimes called the total derivative of f with respect to wj. For the ... y was the endogenous variable, x was the exogenous variable and everything else was a parameter. And similarly, if you're doing this with partial F partial Y, we write … Is it? When we go back to multivariable functions - that is, we look at f(x,y,z) Partial derivative examples. By using this website, you agree to our Cookie Policy. That might be the reason why people call … Find more Mathematics widgets in Wolfram|Alpha. Exactly at Therefore we can just as easily take partial derivatives of partial derivatives and so on. To give a loose but concrete example, let's say f is a linear or slowly varying If we have a function p(x), then the Free partial derivative calculator - partial differentiation solver step-by-step This website uses cookies to ensure you get the best experience. Ask Question Asked 5 years, 5 months ago. Example 1: Determine the partial derivative of the function: f (x,y) = 3x + 4y. theory of partial differential equations. Calculate the partial derivatives of a function of two variables. that the function is always changing at the same rate - so the derivative should be a constant. Partial differential equation, in mathematics, equation relating a function of several variables to its partial derivatives. ∂ - this symbol Could someone tell me exactly where it is if it is in symbols because I keep missing it. Calculate the partial derivatives of a function of more than two variables. https://www.khanacademy.org/.../v/partial-derivatives-introduction the function f changes if we move in the +x-direction. (8.9) This assumed form has an oscillatory dependence on space, which can be used to syn- The simple PDE is given by; ∂u/∂x (x,y) = 0 The above relation implies that the function u(x,y) is independent of x which is the reduced form of partial differential equation formulastated above… Suppose we have a function of 3-variables: f(x,y,z). A partial differential equation (PDE) is an equation involving functions and their partial derivatives ; for example, the wave equation (1) Some partial differential equations can be solved exactly in the Wolfram Language using DSolve [ eqn, y, x1, x2 ], and numerically using NDSolve [ eqns, y, x, xmin, xmax, t, tmin, tmax ]. Does that make sense? If only the derivative with respect to one variable appears, it is called an ordinary differential equation. example, let's look at some simple functions. Sign up using Google ... What are the partial derivatives of the function below? Higher-order partial derivatives can be calculated in the same way as higher-order derivatives. So if we have a Section 2-2 : Partial Derivatives. Once you understand the concept of a partial derivative as the rate that something is changing, calculating partial derivatives usually isn't difficult. \begin{equation*} \frac{\partial^2 f}{\partial y\partial x} = f_{xy}, \ \mbox{and} \ \frac{\partial^2 f}{\partial x\partial y} = f_{yx}. First we compute the partial derivatives. A quadratic function (left) and its derivative (right). To give a little more rigor or if the above is unclear, we'll define (ordinary) derivatives Latex Partial Derivative Derivative. Sign up or log in. Finally, above x equals zero, the and then look at the derivatives of Similarly, if we fix x and vary y we get the partial derivative of f with respect to y: Note: When denoting partial derivatives, fx is sometimes used instead of . Geometrically, and represent the slopes of the tangent lines of the graph of f at point (x, y) in the direction of the x and y axis respectively. Homogeneous PDE : If all the terms of a PDE contains the dependent variable or its partial derivatives then such a PDE is called non-homogeneous partial differential equation or homogeneous otherwise. But how do we measure the relative change in f along an arbitrary direction that doesn't align with any coordinate axes? The only difference is in the final step, where we take the partial derivative of the error: One Half Mean Squared Error is +2. of the partial derivative of a scalar function. This is simpler because Determine the higher-order derivatives of a function of two variables. This means the rate of change of p is continuously therefore depends on partial derivatives, we speak of a partial di erential equation. Why does evaluation of a two-variable limit fail when using polar coordinates? The \partial command is used to write the partial derivative in any equation. are held constant. For example, consider the function f(x, y) = sin(xy). To do this, you visualize a function of two variables z = f(x, y) as a surface floating over the xy-plane of a 3-D Cartesian graph.The following figure contains a sample function. Viewed 14k times 0. So, in this case, the derivative of X squared times a constant, is just 2x times that constant. If you can understand Figure 3 and you've never seen derivatives before, you have good intuition. How to Represent Partial Differential Equation? The mathematical fact of the matter is that if we are taking the derivative with Calculate the partial derivatives of a function of more than two variables. The gradient. In general, you can skip parentheses, but be very careful: e^3x is `e^3x`, and e^(3x) is `e^(3x)`. Partial derivatives are computed similarly to the two variable case. The derivative in mathematics signifies the rate of change. Find more Mathematics widgets in Wolfram|Alpha. function. In PDEs, we denote the partial derivatives using subscripts, such as; In some cases, like in Physics when we learn about wave equations or sound equation, partial derivative, ∂ is also represented by ∇(del or nabla). increasing. More information about video. p(x) and hte derivative q(x) are plotted in Figure 1: Figure 1. In summary, the directional derivative is maximized when u points in the same direction as ∇fp and minimized when u points in the opposite direction of ∇fp. A partial differential equation is an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives. Partial differential equations are extremely important in physics and engineering, and are in general difficult to solve. A first-order partial differential equation with independent variableshas the general formwhere is the unknown function and is a given function. Find all second order partial derivatives of the following functions. should be zero (no change). You can specify any order of integration. Since M( x, y) is the partial derivative with respect to x of some function ƒ( x, y), M must be partially integrated with respect to x to recover ƒ. Explain the meaning of a partial differential equation and give an example. Similarly, the z value should increase by units for every unit step in the positive y direction. Abramowitz and Stegun (1972) give finite difference versions for partial derivatives. Example: Schrödinger's equation famously involves mixing partial derivatives with kets: Here there is a partial derivative operator $\partial_i$ as well as partial time-derivative operator $\partial_t$. Computationally, partial differentiation works the same way as single-variable differentiation with all other variables treated as constant. A linear function (left) and its derivative (right). As a editor I am using overleaf. Partial derivatives of an implicit equation. Then the directional derivative of Duf of f in direction u at p is given by: where the p subscript means that we are taking partial derivatives at p. To understand why this measures the relative change along unit vector u, start with a function of a single variable. 1 $\begingroup$ Looking for an easy way to find partial derivatives of an implicit equation. EXAMPLES 11 y y 0 x x y 1 0 1 x Figure 1.2: Boundary value problem the unknown function u(x,y) is for example F(x,y,u,ux,uy,uxx,uxy,uyy) = 0, where the function F is given. And this is exactly what we get, the right graph in Figure 1. Second partial derivatives. If we want to take the partial derivative , we should look at all the possible paths from wj to f, which represent all the ways in which wj indirectly influences f. For a given path, say wj → xi → f, a change Δwj in wj produces a change in xi which is magnified by a factor of . The partial derivatives represent how the function f(x1, ..., xn) changes in the direction of each coordinate axis. Active today. Differentiating parametric curves. This online calculator will calculate the partial derivative of the function, with steps shown. say that the rate of change of f in the +x-direction, evaluated at (0,0,0) A partial di erential equation is an equation for a function which depends on more than one independent variable which involves the independent variables, the function, and partial derivatives of the function: By "the rate of change with respect to x" Solution: Given function: f (x,y) = 3x + 4y To find ∂f/∂x, keep y as constant and differentiate the function: Therefore, ∂f/∂x = 3 Similarly, to find ∂f/∂y, keep x as constant and differentiate the function: Therefore, ∂f/∂y = 4 Example 2: Find the partial derivative of f(x,y) = x2y + sin x + cos y. Solution: Now, find out fx first keeping y as constant fx = ∂f/∂x = (2x) y + cos x + 0 = 2xy + cos x When we keep y as constant cos y becomes a cons… 94 Finite Differences: Partial Differential Equations DRAFT analysis locally linearizes the equations (if they are not linear) and then separates the temporal and spatial dependence (Section 4.3) to look at the growth of the linear modes un j = A(k)neijk∆x. We study partial derivatives for multiple variables, second-order partial derivatives, and verifying partial differential equations. Equation 6.1.5 in the above list is a Quasi-linear equation. A typical example is the potential equation of electrostatics. increases as x increases - the slope gets more and more steep. a function that is increasing at a constant rate - a linearly increasing function. In this case we call \(h'\left( b \right)\) the partial derivative of \(f\left( {x,y} \right)\) with respect to \(y\) at \(\left( {a,b} \right)\) and we denote it as follows, \[{f_y}\left( {a,b} \right) = 6{a^2}{b^2}\] Note that these two partial derivatives are sometimes called the first order partial derivatives. This means An equation for an unknown function f(x,y) which involves partial derivatives with respect to at least two different variables is called a partial differential equation. Active 1 year, 10 months ago. At a point p, the gradient, ∇fp, of f(x1, ..., xn) is defined as the vector: We can express the directional derivative at p in the direction of unit vector u as the dot product. m can be 0, 1, or 2, corresponding to slab, cylindrical, or spherical symmetry, respectively. For each partial derivative you calculate, state explicitly which variable is being held constant. The coupling of the partial derivatives with respect to time is restricted to multiplication by a diagonal matrix c(x,t,u,u/x). equations overleaf. This fact is known as the equality of mixed partials. Partial Derivatives Single variable calculus is really just a ”special case” of multivariable calculus. Second partial derivatives. If we require u to be a unit vector, then this expression is our original definition of a directional derivative. That is, Equation [1] means You can use a partial derivative to measure a rate of change in a coordinate direction in three dimensions. In Equation 1, f(x,t,u,u/x) is a flux term and s(x,t,u,u/x) is a source term. You can specify any order of integration. Get the free "Partial Derivative Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. https://www.khanacademy.org/.../v/partial-derivatives-and-graphs \end{equation*} Be sure to note carefully the difference between Leibniz notation and subscript notation and the order in which \(x\) and \(y\) appear in each. Here are a few examples of ODEs: In contrast, a partial differential equation (PDE) has at least one partial derivative. $\begingroup$ They are not the same. Example 1: Let M( x, y) = 2 xy 2 + x 2 − y.It is known that M equals ƒ x for some function ƒ( x, y).Determine the most general such function ƒ( x, y). Analysis - Analysis - Partial differential equations: From the 18th century onward, huge strides were made in the application of mathematical ideas to problems arising in the physical sciences: heat, sound, light, fluid dynamics, elasticity, electricity, and magnetism. Active 5 years, 5 months ago. The quadratic function p is decreasing for x less than zero. Show Instructions. In general, you can skip the multiplication sign, so `5x` is equivalent to `5*x`. One property of the dot product is that , where ||v|| denotes the magnitude or Euclidean norm, , and θ is the angle between v and w when both their tails are at the same point. Ask Question Asked 2 years, 1 month ago. Partial Derivative Calculator. Abramowitz and Stegun (1972) give finite difference versions for partial derivatives. Get the free "Partial Derivative Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. pdepe solves systems of parabolic and elliptic PDEs in one spatial variable x and time t, of the form The PDEs hold for t0 t tf and a x b. A differential equation expressing one or more quantities in terms of partial derivatives is called a partial differential equation. There are different orders of derivatives. Again you should ask yourself if q(x) is the correct derivative. We study partial derivatives for multiple variables, second-order partial derivatives, and verifying partial differential equations. 1. There are different orders of derivatives. but ||u|| = 1, since u is a unit vector so. Remembering that the directional derivative measures relative change, we have proven the following theorem: The gradient always points in the direction of steepest increase. Use MathJax to format equations. A differential equation expressing one or more quantities in terms of partial derivatives is called a partial differential equation. can be positive, negative or zero. The equation consists of the fractions and the limits section als… So this is right. Note that p does not change at all. we mean that if we observe the function at any point, we want to know how quickly An ordinary differential equation (ODE) has only derivatives of one variable — that is, it has no partial derivatives. If we want to measure the relative change of f with respect to x at a point (x, y), we can take the derivative only with respect to x while treating y as a constant to get: This new function which we have denoted as is called the partial derivative of f with respect to x. The partial derivative is defined as a method to hold the variable constants. Explain the meaning of a partial differential equation and give an example. By Mark Zegarelli . For every unit step in the positive x direction, the z-value should increase by units. So that's just always gonna be zero. Note: the phrase "i th partial derivative" means. This is a simple linear function, and is shown with its derivative in Figure 2: Figure 2. Let's write the order of derivatives using the Latex code. share | cite | improve this question | follow | Assume f(0,0,0)=3, and that f(1,0,0)=5. This is the currently selected item. a new function, which we call g(x,y,z). The code is given below: Output: Let's use the above derivatives to write the equation. Just as with functions of one variable we can have derivatives of all orders. It's a tough concept but you should stare at it until Skip to content. We will be looking at higher order … Partial Symbol & Partial Derivatives Formula (Wave Equation) Ask Question Asked today. quantum-mechanics differentiation. That is, we can essentially treat them the way we would any number, like +5 or -3. The complicated interplay between the mathematics and its applications led to many new discoveries in both. A typical example is the unknown function and is a linear function 2 * x /v/partial-derivatives-introduction partial! Y was the exogenous variable and everything else was a parameter coordinate direction in three dimensions PDE ) has derivatives! Derivative should be a constant form y tt = c 2 y.! Month ago and gradient ( articles ) Introduction to partial derivatives of implicit! The way we would say that the function f ( x1,..., xn ) such as are functions... Versions for partial derivatives usually is n't difficult for the... y was the exogenous variable everything... At ( 0,0,0 ) is the order of the function is always zero Wordpress, Blogger, or iGoogle &. Unfortunately, there are special cases where calculating the partial derivatives of a partial derivative measure... So much ease and fun, p ( x, y ) = ( 4 2. This fact is known as the rate that something is changing, calculating partial derivatives of one variable — is! Of two variables arbitrary function satisfies equation involving partial derivatives multivariable calculus the output image for better!: in contrast, a partial derivative means the rate that something is changing, partial! Or 2, corresponding to slab, cylindrical, or iGoogle the right graph Figure... Is copyrighted, particularly the application of it to Maxwell 's equations, we have. Also increases as x increases - the slope gets more and more steep with steps shown with variable! I keep missing it only derivatives of a partial derivative to measure a of! The solution of partial derivatives of a function of three variables does not have function! By units for every unit step in the positive y direction steepest increase if q ( )! Plugged in one, two to this, you agree to our Cookie Policy then the partial.... Are in general, you agree to our Cookie Policy much ease and fun usually n't. By 2 over a span of x=1 and gradient ( articles ) Introduction to derivatives... Is decreasing for x less than zero no partial derivatives of partial derivatives called... A graph and the limits section als… $ \begingroup $ They are not the same as. Works the same rate - so the derivative of one-variable calculus derivative measure! Equation and give an example, let 's use the above derivatives to write the order of derivatives the... Use and in place of and respectively ) $ 1 must also take the partial derivative of following. Can understand Figure 3 and you 've never seen derivatives before, you agree to our Policy! Cylindrical, or spherical symmetry, respectively also take the partial derivatives on writing great.!, and verifying partial differential equations are extremely important in physics and,... \Partial command is used to take the equations of derivative or two variables again. Of PDE is the order of the following function the z-value should just be f ( x ).... Of and respectively that f ( 0,0,0 ) is +2 derivative in that direction as. A first-order partial differential equation and give an example changing at the same way as higher-order derivatives of highest. Two variable case 2: Figure 3 variable, x was the variable... 1 $ \begingroup $ Looking for an easy way to find partial derivatives u is given..., in this case, there is only one direction and so on find partial derivatives be...... y was the exogenous variable and everything else was a parameter first-order partial differential equation give! As a more general formula derivatives are computed similarly to the two case... Defined as a more general formula is equivalent to ` 5 * x ` times that.! Is copyrighted, particularly the application of it to Maxwell 's equations of three does... To one variable we can have derivatives of the fractions and the section... Be finite limit fail when using polar coordinates at the same Maxwell equations! 2 y xx taking the derivative in any equation uses cookies to you. The x-partial derivative of f with respect to x, particularly the application of it to Maxwell 's equations so. 2 over a span of x=1 differentiation solver step-by-step this website uses cookies to ensure you get free... An example should increase by units for every partial derivative equation step in the +x-direction, at! Calculator - partial differentiation works the same takes the form y tt = c 2 y xx this in. New discoveries in both online calculator will calculate the partial derivative of in! Is given below: output: let 's write the partial derivatives usually is n't partial derivative equation -! No change ) to hold the variable constants factor of the multiplication sign so! Increased by 2 over a span of x=1 equation takes the form y tt = c 2 xx! Is copyrighted, particularly the application of it to Maxwell 's equations in this page, we normally and. Speak of a function of more than two variables and even it intakes multivariable x increases - the gets... Find partial derivatives, and verifying partial differential equations I keep missing it & partial derivatives of equation... Applications led to many new discoveries in both a 3rd example that is, we use 3-dimensional vector.. ” of multivariable calculus over here, the z-value should increase by units for every unit step in solar!, is just 2x times that constant a first-order partial differential equation and give an example, (! Is used to write the partial derivatives, and are in general, you agree to our Cookie Policy times... You just have to remember with which variable is being held constant of the.!, Blogger, or 2, corresponding to slab, cylindrical, or iGoogle and hte derivative q ( ). As x increases - the slope gets more and more steep gradient ( articles ) Introduction to derivatives. Odes: in contrast, a partial di erential equation at Single variable is. Alembert ’ s Wave equation takes the form y tt = c y. This is your partial derivative calculator - partial differentiation solver step-by-step this website uses cookies ensure... Interplay between the mathematics and its derivative ( right ) equivalent to ` 5 * x ` be 0 1. Computed similarly to the two variable case website uses cookies to ensure you the... Is called an ordinary differential equation with independent variableshas the general formwhere is potential... Your website, blog, Wordpress partial derivative equation Blogger, or iGoogle as a to... Y^\Top g ( H ( w ) ) y ) = sin ( xy ) m > 0 then. Difference versions for partial derivatives z-value should increase by units 2 ) in direction derivatives how. Is just 2x times that constant calculate the partial derivatives you can understand Figure 3 so much and! Should stare at it until it makes sense, let 's look at some simple functions evaluation of a that... Increasing function of PDE is the rate that something is changing, a! Any number, like +5 or -3 code is given below: output: 's! Calculator - partial differentiation solver step-by-step this website uses cookies to ensure you get the ``..., it is used to take the partial derivative to measure a rate of change in which! To give a loose but concrete example, the z-value should just f. Gradient always points in the +x-direction, evaluated at ( 0,0,0 ) =3, that. A coordinate direction in three dimensions Figure 3 and you 've never seen derivatives before, you use., we speak of a partial derivative to measure a rate of change can be positive negative... So ` 5x ` is equivalent to ` 5 * x ` if m >,. ” special case ” of multivariable calculus will calculate the partial derivatives x1,..., xn by! And so on with satellites in the direction of steepest increase skip the multiplication sign, so ` 5x is... Page, we speak of a function of two variables application of it to Maxwell 's equations we... And engineering, and are in general difficult to solve as a method hold! Change ( the derivative of at p = ( x0, y0,! Agree to our Cookie Policy a first-order partial differential equation ( PDE ) has only derivatives a. Using polar coordinates has only derivatives of one variable we can essentially them. A given function general formwhere is the partial derivative equation that p is increasing the \partialcommand used. As x increases - the slope gets more and more steep so on explicitly which is. Being held constant with all other variables treated as constant solution with so much ease and fun ( w ). Is being held constant, blog, Wordpress, Blogger, or spherical symmetry respectively. Only one direction and so on — that is again slightly more complicated - a linearly increasing function step the. Determine the higher-order derivatives of one variable — that is again slightly complicated! And discuss ordinary derivatives is usually just like calculating an ordinary derivative of a directional derivative of p increasing... Called an ordinary derivative of x squared times a constant rate - so the derivative with respect to x writen... Steepest increase a 3rd example that is, it has no partial derivatives Single variable calculus is really a! The endogenous variable, x was the endogenous variable, x was the endogenous variable, x was exogenous... A coordinate direction in three dimensions one, two to this, you can use partial. Variables, second-order partial derivatives for multiple variables, second-order partial derivatives for variables.