Question

Given k(x)=f(g(x)) and
f(2)=-4 g(2)=2
f'(2)=3 g'(2)=5
Find k(2) and k'(2)

Answer 1

\(\Large\bf \color{#CC0033 }{\text{Welcome to OpenStudy! :)}}\)

Answer 2

\[\Large\bf\sf k(x)=f(g(x))\]By the chain rule:\[\Large\bf\sf k'(x)=f'(g(x))\cdot g'(x)\]

Answer 3

\[\Large\bf\sf k'(2)=f'(g(2))\cdot g'(2)\]

Answer 4

Then we just use the information they provided us with to plug in the pieces.

Answer 5

That helps a little.
Now I get:
k(2) = -4
and
k'(2) = - 8

How does that look?

Answer 6

Your k(2) looks correct:\[\Large\bf\sf \color{royalblue}{g(2)=2}\]\[\Large\bf\sf f(2)=-4\]
\[\Large\bf\sf k(2)\quad=\quad f\left[\color{royalblue}{g(2)}\right]\quad=\quad f\left[\color{royalblue}{2}\right]\quad=\quad -4\]

Answer 7

\[\Large\bf\sf k'(2)\quad=\quad f'\left[\color{royalblue}{g(2)}\right]\cdot g'(2)\]
\[\Large\bf\sf =\quad f'\left[\color{royalblue}{2}\right]\cdot g'(2)\quad=\quad 3\cdot 5\]