From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!
We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.
© 2016 Shmoop University, Inc. All rights reserved.

The Chain Rule in Leibniz Notation Examples

Example 1

Let z = y4 and y = 3x + 5. What is ?

Example 2

Let z = cos(sin x). Find the derivative of z with respect to x.

Example 3

If we have

q = f(r) and r = g(i), what does the chain rule say?

Example 4

Let u = 5z + z2 and let y = ln u. Find 

Example 5

Let q = sin(6s + s-1). Find .