Question

Using Chain Rule: Write the composite function in the for f(g(x)). [Identify the inner function u=g(x) and the outer function y=f(u).] Then find the derivative for y= e^ sqrt of x

Answer 1

Oh, you again :P hah!

Answer 2

Lol I'm sorry, I just suck at AP Calc

Answer 3

These aren't as difficult to break up as you might think.
I would do it like this:\[\Large\rm f=e^{stuff}\]\[\Large\rm g=stuff\]

Answer 4

\[\Large\rm u=g(x)=\sqrt x\]\[\Large\rm y=f(u)=e^u\]Yah? :d

Answer 5

I'm lost as to how you would set the first part of the equation up...

Answer 6

Ummmmm let's see...

Answer 7

\[\Large\rm \color{orangered}{g(x)=\sqrt x}\]We want to "stuff this into" f(x).
Think of f(x) as an outer shell that contains some stuff.

In the last problem our outer shell was ( )^2
and we stuffed g(x), or 1-x^2, into that, yes?

So in this problem we want to think of e^( ) as our outer shell,
and we want to stuff g(x) into that.

Answer 8

\[\Large\rm f(~~~)=e^{(~~~)}\]

Answer 9

We take our g(x) and stuff it into our f function,
\[\Large\rm f\left(\color{orangered}{g(x)}\right)=e^{(\color{orangered}{g(x)})}\]And on the right side, we replace g(x) with what it actually is,\[\Large\rm f\left(\color{orangered}{g(x)}\right)=e^{(\color{orangered}{\sqrt x})}\]

Answer 10

brb

Answer 11

So wouldn't you change the sqrt of x to x^1/2? to continue the problem or would it remain as is?
ok

Answer 12

You can change sqrt(x) to x^(1/2) if you find that helpful.
It will make it easier to apply your chain rule, yes.

sqrt(x) comes up so often though, that it's really worth memorizing the derivative if you can squeeze it into your noggin.
`The derivative of square root x is 1 over 2 square roots of x.`\[\Large\rm \left(\sqrt{x}\right)'=\frac{1}{2\sqrt x}\]

Answer 13

But yes, if you prefer, you can write it with a rational exponent as apply your power rule :)\[\Large\rm \left(x^{1/2}\right)'=\frac{1}{2}x^{-1/2}\]