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Let $f'(x) = \sqrt{3x + 4}$ and $g(x)=x^2-1$. Find $h'(x)$, if $h(x) = f(g(x))$.

I know that $g'(x) = 2x$, but I don't know how to do further. The answer is $h'(x) = 2x \sqrt{3x^2 + 1}$.

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1  
I think you want to find $fog^{'}(x)$, so you have to use this formula $$fog^{'}(x)=f^{'}(g(x).g^{'}(x)$$. –  Kns Apr 22 '12 at 14:09
    
@Sb Sangpi ,rule for derivate composite function is following $(f(g(x))'=f'(g(x))*g'(x)$ –  dato datuashvili Apr 22 '12 at 14:18

3 Answers 3

up vote 2 down vote accepted

You should be asking "find $h'(x)$ if $h(x)=f\bigl(g(x)\bigr)$".

Use the chain rule: $$\bigl[\,f\bigl(g(x)\bigr)\,\bigr]' = \color{maroon}{f'\bigl(g(x)\bigr)}\cdot\color{darkgreen}{ g'(x)}.$$

We aren't told what $f$ is, but we do know that $$f'(\color{darkblue}{x})=\sqrt{\,3\color{darkblue}{x}+4}.$$ Then $$\color{maroon}{f'\bigl(g(x)\bigr)}=f'(\color{darkblue}{x^2-1})=\sqrt{ 3(\color{darkblue}{x^2-1})+4}=\color{maroon}{\sqrt{3x^2+1}}.$$
Also $\color{darkgreen}{g'(x)}=(x^2-1)'=\color{darkgreen}{2x}$.

So, using the chain rule:

$$ \bigl[\,f\bigl(g(x)\bigr)\,\bigr]' =\color{maroon}{ f'\bigl( g(x) \bigr)} \cdot \color{darkgreen}{g'(x) }= \color{maroon}{\sqrt{3x^2+1 }} \cdot\color{darkgreen}{2x}=2x\sqrt{3x^2+1}. $$

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Hint:

The general rule for calculating the derivative of a composite functions is: $$(g(f(x)))'=g'(f(x))\cdot f'(x)$$

For example, let $f(x)=x^2$ and $g(x)=\sin(x)$. Then $f'(x)=2x$ and $g'(x)=\cos(x)$. For the composite function $g(f(x))$ you get: $$(g(f(x)))'=g'(f(x))\cdot f'(x)=\cos(f(x))\cdot 2x=\cos(x^2)\cdot 2x$$

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if f'(x) is given,then we are you trying to find f'(x)? –  dato datuashvili Apr 22 '12 at 14:06
    
The derivative of the normal function f'(x) is given. Sb Sangpi is trying to find the derivative of a composite function, that has the form f(g(x)) –  chemeng Apr 22 '12 at 14:09

$h'(x)=f'(g(x))*g'(x)$

$g'(x)=2x$ as you said

Now what is $f'(g(x))$ equal to? $f'(g(x))=\sqrt{(3(x^2-1)+4)}$ which is equal to $\sqrt{3x^2+1}$. Now if you multiply, you will get the desired result.

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