Section 4.3 - Partial Derivatives

Section Objectives

1. Compute the partial derivatives of a function of several variables.
2. Compute higher-order partial derivatives.
3. Describe the conditions for equality of higher-order mixed partial derivatives.
4. Interpret partial derivatives as slopes.

Partial derivatives

Perhaps the simplest way to generalize the Calculus I derivative to functions of several variables is to apply the change independently to each variable. This gives us the partial derivatives:

and

provided the limits exist.

It should be clear that each partial derivative treats the other variable as non-changing or constant. There are a few important ideas we can take away from the definitions:

1. The partial derivatives represent slopes. For example, $f_x(x,y)$$f_x(x,y)$ is the slope of the surface at the point $(x,y)$$(x,y)$ in the direction of the positive $x$$x$-axis.
2. Partial derivatives can be evaluated by using our Calculus I derivative rules. We simply treat the other variable as a constant.
3. Higher-order derivatives are also defined by treating variables as constant.

Examples

• Let $f(x,y)=x^2-3xy+2y^2-4x+5y-12$$f(x,y)=x^2-3xy+2y^2-4x+5y-12$. Determine $\partial f/\partial x$$\partial f / \partial x$ and $\partial f/\partial y$$\partial f / \partial y$.

• Let $g(x,y)=\sin (x^{2}y-2x+4)$$g(x,y)=\sin(x^2y-2x+4)$. Determine $\partial g/\partial x$$\partial g / \partial x$ and $\partial g/\partial y$$\partial g / \partial y$.

• Find the slope of the graph of $z=\sqrt{9-x^2+y^2}$$z=\sqrt{9-x^2+y^2}$ at the point $(3,4)$$(3,4)$ in the direction of the positive $x$$x$-axis.

• Let $F(x,y,z)=e^{x{y}^2{z}^3}$$F(x,y,z)=e^{xy^2z^3}$. Determine $\partial F/\partial x$$\partial F / \partial x$, $\partial F/\partial y$$\partial F / \partial y$, and $\partial F/\partial z$$\partial F / \partial z$.

• Let $f(x,y)=x^2-3xy+2y^2-4x+5y-12$$f(x,y)=x^2-3xy+2y^2-4x+5y-12$. Determine $f_{xy}$$f_{xy}$ and $f_{yx}$$f_{yx}$.

Equality of mixed partials

From the last example, you may expect that mixed partial derivatives must be equal. They don't have to be! But they usually are.

Theorem: Suppose that $f(x,y)$$f(x,y)$ is defined on an open region $R$$R$. If $f_{xy}$$f_{xy}$ and $f_{yx}$$f_{yx}$ are continuous on $R$$R$, then $f_{xy}=f_{yx}$$f_{xy}=f_{yx}$ on $R$$R$.

Examples

• Let $g(x,y)=x{e}^{-3y}+\sin (2x-5y)$$g(x,y)=xe^{-3y} + \sin(2x-5y)$. Compute all four 2nd-order partial derivatives.

• Let $H(x,y,z)=5x^2-7y^2+z^2-2xz+8xyz-13z$$H(x,y,z) = 5x^2-7y^2 + z^2 - 2xz + 8xyz - 13z$. Compute $H_{zyx}$$H_{zyx}$ and $H_{xyz}$$H_{xyz}$.