Partial Derivatives

≈ 25 min

A function of two variables $f(x, y)$ has a rate of change in each input direction separately. The partial derivative $\dfrac{\partial f}{\partial x}$ is found by differentiating with respect to $x$ while holding $y$ constant (treat every $y$ as if it were a fixed number), and $\dfrac{\partial f}{\partial y}$ holds $x$ constant.

Worked example. For $f(x, y) = x^2 y + 3xy^3$ :

\frac{\partial f}{\partial x} = 2xy + 3y^3, \qquad \frac{\partial f}{\partial y} = x^2 + 9xy^2.

Second-order partials repeat the process. The mixed partial $f_{xy} = \dfrac{\partial}{\partial y}\left(\dfrac{\partial f}{\partial x}\right)$ .

For $f(x, y) = 3x^2 y + 2y^3$ , evaluate $\dfrac{\partial f}{\partial x}$ at the point $(1, 2)$ .

For $f(x, y) = x^2 y + 3xy^3$ , find $\dfrac{\partial f}{\partial x}$ . Write your answer as an expression in $x$ and $y$ .

What is $\dfrac{\partial}{\partial x}\big[\sin(xy)\big]$ ?

$y\cos(xy)$

$x\cos(xy)$

$\cos(xy)$

$-y\cos(xy)$

For $f(x, y) = x^2 y^3$ , find the mixed partial derivative $f_{xy}$ . Write your answer as an expression in $x$ and $y$ .