1. ## Differentiating given f(x)

Given y=f(x) with f(1)=4 and f'(1)=3, find:
A) g'(1) if g(x) = √(f(x))
B) h'(1) if h(x) = f(√(x))
And show how you got there so I can learn.

2. ## Re: Differentiating given f(x)

What have you tried? What is $g'(x)$ and $h'(x)$?

3. ## Re: Differentiating given f(x)

These are both about using the chain rule.
I'll use a function other than the square root to give you an idea.

$\text{A) Suppose } h(x) = [f(x)]^3$

$\text{Then } h'(x) = 3[f(x)]^2f'(x)$

$\text{so } h'(1) = 3[f(1)]^2f'(1) = 3[4]^2(3) = (9)(16) = 144.$

$\text{B) Suppose } \alpha(x) = f(x^3)$

$\text{Then } \alpha'(x) = f'(x^3)(3x^2)$

$\text{so } \alpha'(1) = f'(1^3)(3(1^2)) = 3f'(1) = (3)(3) = 9.$

4. ## Re: Differentiating given f(x)

Originally Posted by johnsomeone
These are both about using the chain rule.
I'll use a function other than the square root to give you an idea.

$\text{A) Suppose } h(x) = [f(x)]^3$

$\text{Then } h'(x) = 3[f(x)]^2f'(x)$

$\text{so } h'(1) = 3[f(1)]^2f'(1) = 3[4]^2(3) = (9)(16) = 144.$

$\text{B) Suppose } \alpha(x) = f(x^3)$

$\text{Then } \alpha'(x) = f'(x^3)(3x^2)$

$\text{so } \alpha'(1) = f'(1^3)(3(1^2)) = 3f'(1) = (3)(3) = 9.$
Even better answer than I was expecting, now I can do it (and future problems) on my own, thank you.