The Partial Derivative

The Partial Derivative is a way to think about the derivative of a function of multiple variables.

Table of Contents

• Table of Contents
• Introduction
• Example Problem: Evaluating a Simple Partial Derivative
• Formalizing Partial Derivatives
• Partial Derivatives in Cylindrical Coordinates (Optional)
• Summary and Next Steps

Introduction

Recall that one way to conceptualize the derivative is to think: "If I nudge the input a little bit, how much does the output change?" So, for , you would change by a small amount and see how much changes.

For functions with multiple inputs, what we can do is nudge each input separately and see how much the output changes. So, for example, if we have a function , we can nudge by a small amount and see how much changes. This means that we treat as a constant while we nudge .

This is known as the partial derivative of with respect to , denoted in these ways:

• Leibniz notation:
• Lagrange notation:
• Euler notation:

It's called a "partial" derivative because we're only considering one input at a time, so it's not a "complete" way to measure the rate of change.

Example Problem: Evaluating a Simple Partial Derivative

is defined as follows:

Find .

We want to evaluate the rate of change of with respect to when and . So, keeping constant at , we can differentiate with respect to :

So, .

Formalizing Partial Derivatives

We can formally define the partial derivative so that it's mathematically rigorous. Recall the definition of the derivative for a single-variable function:

This is essentially a fancy way of calculating the rise over run.

Look at the term - this is the output when we nudge the input by . For a function of multiple variables, we can nudge each input separately. So, given a function , if we want to find the partial derivative with respect to , we nudge by and keep constant; .

Everything else in the definition of the derivative remains the same. So, the partial derivative of with respect to is defined as:

Visualizing Partial Derivatives

Let's visualize the concept of partial derivatives with a simple example. Consider the function .

Suppose we want to find the partial derivative of with respect to at the point . This means that is kept constant at while we nudge . One way to think of this on a graph is to imagine a slice of the function where .

So first, here's the graph of :

If we slice the function at , we get a parabola:

Then, the nudge in can be represented by an arrow on the graph. We can compute the partial derivative by finding the derivative of this slice at the point :

Likewise, we can slice the function in the -direction to find the partial derivative with respect to :

The partial derivative with respect to at is:

Partial Derivatives in Cylindrical Coordinates (Optional)

In cylindrical coordinates, we have three variables: , , and .

The and variables are essentially polar coordinates that lie on the -plane, and stays the same.

So, if we have a function , we can find the partial derivatives with respect to and by treating as a constant. For example, the partial derivative with respect to is:

This is similar to the partial derivatives in Cartesian coordinates, but we're nudging instead of .

Summary and Next Steps

We have introduced the concept of partial derivatives and some ways to visualize them.

Here are the key points to remember:

1. The partial derivative of a function with respect to one of its inputs measures the rate of change of the function when that input is nudged.

2. The partial derivative is denoted by , where is the input we are differentiating with respect to.

3. The partial derivative is calculated by treating all other inputs as constants and finding the derivative as usual. The formal definition is:

4. Partial derivatives can be visualized by slicing the function in the direction of the input we are differentiating with respect to.

Next, we shall consider second-order partial derivatives, similar to how we have second derivatives for single-variable functions.