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