Online Test Banks
Score higher
See Online Test Banks
eLearning
Learning anything is easy
Browse Online Courses
Mobile Apps
Learning on the go
Explore Mobile Apps
Dummies Store
Shop for books and more
Start Shopping

How to Use the Chain Rule to Find the Derivative of Nested Functions

Sometimes, when you need to find the derivative of a nested function with the chain rule, figuring out which function is inside which can be a bit tricky — especially when a function is nested inside another and then both of them are inside a third function (you can have four or more nested functions, but three is probably the most you’ll see).

image0.png

Now it’s easy to see the order in which the functions are nested. The innermost function is inside the innermost parentheses — that’s

image1.png

Next, the sine function is inside the next set of parentheses — that’s

sin(stuff)

Last, the cubing function is on the outside of everything — that’s stuff3

image2.png

Okay, now that you know the order of the functions, you can differentiate from outside in.

  1. The outermost function is stuff cubed and its derivative is given by the power rule.

    image3.png
  2. As with all chain rule problems, you multiply that by stuff'.

    image4.png
  3. Put the stuff,

    image5.png

    back where it belongs.

    image6.png
  4. Use the chain rule again.

    image7.png
  5. The stuff is

    image8.png

    and its derivative is 10x – 4. Plug those things back in.

    image9.png
  6. Now that you’ve got the derivative of

    image10.png

    plug this result into the result from Step 3, which gives you the whole enchilada.

    image11.png
  7. This can be simplified a bit.

    image12.png

It may have occurred to you that you can save some time by not switching to the word stuff and then switching back. That’s true, but the technique forces you to leave the stuff alone during each step of a problem. That’s the critical point. Make sure you DO NOT TOUCH THE STUFF.

  • Add a Comment
  • Print
  • Share
blog comments powered by Disqus
Advertisement

Inside Dummies.com

Dummies.com Sweepstakes

Win $500. Easy.