Question

Suppose the functions f and g and their derivatives have the following values at x=1 and x=2. Let h(x) = f(g(x)), Evaluate h'(1).

Follow steps of video:https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-1b/v/chain-rule-example-implicit

Answer should be -6pi

Answer 1

Answer:

$h'(1)=-6\pi$

Step-by-step explanation:

Given:

The values of $f(x),f'(x),g(x)\ and\ g'(x)$ are given as:

$f(1)=8,f(2)=3\\\\g(1)=2,g(2)=-4\\\\f'(1)=\frac{1}{3},f'(2)=2\pi\\\\g'(1)=-3,g'(2)=5$

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

Differentiating both sides with respect to 'x'. This gives,

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

Now, in order to find $h'(1)$, plug in 1 for 'x'. This gives,

$h'(1)=f'(g(1))\times g'(1)$

Now, plug in $g(1)=2,g'(1)=-3$

$h'(1)=f'(2)\times -3$

Now, plug in $f'(2)=2\pi$

$h'(1)=2\pi\times -3\\\\h'(1)=-6\pi$

Hence, the value of $h'(1)$ is $-6\pi$.