If $h(x)=f(g(x))$, find $h'(9)$

Key concept: This is a standard composition derivative problem. The derivative of a composite function uses the chain rule, then the result is evaluated at the requested point.

Step by step

Step 1: Recognize the composition

The function is

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

Step 2: Differentiate using the chain rule

$h'(x)=f'(g(x))\cdot g'(x)$

Step 3: Evaluate at $x=9$

$h'(9)=f'(g(9))\cdot g'(9)$

Final answer

$h'(9)=f'(g(9))\cdot g'(9)$

Pitfall alert

A common error is writing only $f'(x)g'(x)$ and forgetting that the outside derivative must be evaluated at the inside function, $g(x)$.

Try different conditions

If the composite were $h(x)=f(g(9))$ with no $x$ inside, then $h(x)$ would be a constant and the derivative would be $0$.

Further reading

composition, chain rule, derivative of composite function