finding derivative with fundamental theorem of calculus chain rule

[wind solar]

I need help with figuring out how to find derivatives with chain rule. No specific question, just need help.

[indigo kettle]

It’s just the plug and chug formula $d/dx [f(g(x)) = f’(g(x))*g(x)$

[royal drift]

CJ_:)

[wind solar]

Oh, thanks

[vivid summit]

Here's how to memorize it
First write clearly the function $g$ inside the function $f$:

$ h(x) = f(g(x))$
We want to find the derivative of $h$.

Let's take an example where we want the derivative of $h(x) = (x^2+1)^3$

So we write
$h(x) = f(g(x))$ with
$g(x) = x^2+1$ and $f(x)=x^3$

[royal drift]

Daddy_314

[vivid summit]

Now, the formula says
" the derivative of h is the derivative of g (the inside function) times the derivative of f taken at g(x) "

In other words, we take the derivative of $g$
$g'(x) = 2x$
The derivative of $f$ is
$f'(x) = 3x^2$

[royal drift]

Daddy_314

[vivid summit]

And then we evaluate f'(x) at g(x)

Stitching all together we get:
$h'(x) = g'(x) \times f'[g(x)]$

[royal drift]

Daddy_314

[vivid summit]

$h'(x) = 2x \times 3(x^2+1)^2$

[royal drift]

Daddy_314

[vivid summit]

Therefore $h'(x) = 6x(x^2+1)^2$

[royal drift]

Daddy_314