13. Partial Derivatives

Imagine we are hiking and are standing at the blue dot on this contour map. What is the slope at that point? Or, how steep is the mountain at that point? This was a straightforward question in single variable calculus, but it is not straightforward now. There are 360 degrees of directions to choose from. How steep is if we're facing north? East? How about 14 degrees west of south?

Today we will begin to answer those questions with the introduction of multivariable derivatives. We start with just two of the $\large 360^{\circ}$ of directions. The two primary directions to consider are the $\large x$ and $\large y$ directions, and the slopes in these directions are called partial derivatives.

The partial derivative of $\large f(x,y)$ with respect to $\large x$ is the slope of $\large f(x,y)$ in the $\large x$-direction. We find this by simply holding $\large y$ constant and by taking the derivative of the function with respect to $\large x$ only. Here's an illustration of the idea:

From: https://mathinsight.org/partial_derivative_limit_definition

The actual definition, not surprisingly, involves limits:

$$\large \begin{aligned} f_x(x,y) &= \lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}\\\ f_y(x,y) &= \lim_{h\to 0}\frac{f(x,y+h)-f(x,y)}{h} \end{aligned}$$

We see in the definition for the partial derivative with respect to $\large x$, the $\large y$ value is held constant while we look at a $\large \Delta h$ change in the $\large x$ value only.

Notation

There are two notations for partial derivatives:

$$\large \begin{aligned} f_x(x,y)\text{ and }f_y(x,y)\\\ \frac{\partial f}{\partial x}(x,y)\text{ and }\frac{\partial f}{\partial y}(x,y) \end{aligned}$$

We will typically use the subscript notation, but the Leibniz notation is sometimes useful as well. We should be familiar with both. The symbol $\large \partial$ is just called a "partial" and is a symbol invented for partial derivatives (not a Greek letter or other symbol).

Computing Partial Derivatives

To compute a partial derivative, we just treat the other variable as if it were a constant, and we take the derivative as if we had a single variable function:

We can find higher order partial derivative by repeating the process. Let $\large f(x,y)=2xy^2+y\cos(x)$. Then:

$$\large \begin{split} f_x(x,y)&=2y^2-y\sin(x)\Rightarrow f_{xx}=-y\cos(x)\\\ f_y(x,y)&=4xy+\cos(x)\Rightarrow f_{yy}=4x\\\ \end{split}$$

We can also compute mixed partial derivatives. In the example above:

$$\large \begin{split} f_x(x,y)&=2y^2-y\sin(x)\Rightarrow f_{xy}=4y-\sin(x)\\\ f_y(x,y)&=4xy+\cos(x)\Rightarrow f_{yx}=4y-\sin(x)\\\ \end{split}$$

Notice in the example that $\large f_{xy}=f_{yx}$. This is true most of the time (whenever the mixed partials are continuous functions)!

Tomorrow's class:

• Practice computing partial derivatives
• A physical example

Section Outcomes:

I can:

• Compute partial derivatives
• Interpret partial derivatives as slopes and rates of change

Other resources

For more, see https://openstax.org/books/calculus-volume-3/pages/4-3-partial-derivatives