Fred has mastered differentiating when a function has only one variable. Somehow, he missed the lessons on partial differentiation. He’s starting to panic because the current homework deals with higher-order partial derivatives.

In this lesson, we review partial differentiation by using examples. Then, we extend the method to higher-order partial derivatives. Let’s zoom in on Fred to see what he’s currently trying to figure out.

Our fearless math student, Fred, is being asked to find the derivative of x⁴ – x³. He correctly answers: 4x³– 3x². Let’s help Fred with his review of partial derivatives. You ask him to find the derivative of x⁴a – x³b where a and b are constants. Again, a correct answer: 4x³a– 3x²b.

Now the fun begins. Let’s read the following math statement:

It says: the partial (derivative) with respect to x. The word ”derivative” is in parentheses because we often just say: the partial with respect to x.

Let’s write the derivative of x⁴a – x³b as a partial derivative:

The answer is still 4x³a– 3x²b.

The a and the b are constants. When taking the partial derivative with respect to x, everything other than x is a constant. So we could write:

and y⁶ and y⁵ are treated as constants. So, the answer is still the same except the a and b constants are more specific. The answer is now x⁴y⁶ – x³y⁵. Thus, if f(x, y) is the function:

and y⁶ and y⁵ are treated as constants

Of course, we can also have a partial with respect to y where everything else is constant. To do these kinds of derivatives, you develop a selective focus with your eyes. You ”see” the y as the variable and the x⁴ and the x³ as constants.

Thus,

And this is pretty much the idea with partial derivatives. You just have to be careful with the ”with respect to” part. Fred is ready to move on to higher-order derivatives.

Second-Order Partials

Let’s say we want to take the partial derivative of the partial derivative. We write this as:

and read this as the second partial with respect to x. This is called a second-order partial derivative. Using the example for f(x, y), here are the details:

We have:

• Line 1: The partial squared with respect to x squared of f with respect to x squared of f is the partial with respect to x of the partial of f with respect to x. We have already computed the quantity in parentheses.
• Line 2: Replace the quantity in parentheses with 4x³y⁶ – 3x²y⁵.
• Line 3: Take the derivative with respect to x gives 12x²y⁶ – 6xy⁵.

For practice, we ask Fred to work on the second partial with respect to y of f. The details:

This idea of the partial derivative of the partial derivative of the … can be extended to even higher-order derivatives.

Fred is overjoyed at his success with higher-order partial derivatives. But wait, there’s one more item to check out.