Question

A table of values for f, g, f', and g' is given

x f(x) g(x) f'(x) g'(x)
1 2 4 3 -7
2 4 1 -5 8
4 1 2 6 -9
If h(x) = g(f(x^2)), find h'(2)

The answer is -168. I've tried everything. No matter how far I go in using chain rule, I keep getting different answers and none of them are -168. Help!

Answer 1

As you've mentioned,
`h'(2) = -168`. See explanation.

Explanation:

`h(x) = g(f(x^(2)))` is a composite function about
`x`.

`\displaystyle \begin{aligned}h'(x) &= (d)/(dx)[g(f(x^(2)))] \\ &=g'(f(x^(2)) \cdot (d)/(dx)[f(x^(2))]&&\text{Chain rule; treat} \; f(x^(2))\;\text{as the inner function.} \\&=g'(f(x)^(2))\cdot f'(x^(2))\cdot (d)/(dx)[x^(2)] &&\text{Chain rule; treat} \; x^(2)\;\text{as the inner function.} \\ &= g'(f(x^(2)))\cdot f'(x^(2)) \cdot 2\;x&&\text{Power Rule.}\; (d)/(dx)[x^(2)] = 2\;x. \\&= 2\;x \cdot f'(x^(2))\cdot g'(f(x^(2)))\end{aligned}`.

For
`x = 2`:

• 
  `2\;x = 4`;
• 
  `x^(2) = 4`;
• 
  `f'(x^(2)) = f(4) = 6`;
• 
  `f(x^(2)) = 1`;
• 
  `g'(f(x^(2)) = g'(1) =-7`.

`\displaystyle h'(x) = 2\;x \cdot f'(x^(2))\cdot g'(f(x^(2))) = -168`.