# How do you find the derivative of #f(x)= cos (sin (4x))#?

##### 2 Answers

#### Explanation:

#"differentiate using the "color(blue)"chain rule"#

#"given "y=f(g(h(x)))" then "#

#dy/dx=f'(g(h(x))xxg'(h(x))xxh'(x)#

#y=cos(sin(4x))#

#dy/dx=sin(sin(4x))xxd/dx(sin(4x))xxd/dx(4x)#

#color(white)(dy/dx)=sin(sin(4x))xx-cos(4x)xx4#

#color(white)(dy/dx)=-4cos(4x)sin(sin(4x))#

Rewrite

#### Explanation:

I'd break up the function composition into several parts. Here we have

- It multiplies the input by
#4# , - It takes the sine value of the result from the above,
- It takes the cosine value of the result from the above.

If we have

Which should be doing each step from the inside, out. Then we can use the chain rule, which to me is more of a method than a rule. Here's how it goes:

Start from the innermost layer, taking the derivative of

We also have:

What we did was solve for

Solving for a tiny nudge in

Then, we evaluate

Now, the tiny nudge is no longer in terms of

And so is the derivative! What we have left to do is

Then we'll "unroll" things, first evaluating

Then evaluating

Simplifying to make things look neater:

And finally "divide" by

This is not only the derivative of