Let r(x)= f(g(h(x))), where h(1)=4, g(4)=3, h'(1)=3, g'(4)=3, f'(3)=8. Find r'(1).

Question

Let r(x)= f(g(h(x))), where h(1)=4, g(4)=3, h'(1)=3, g'(4)=3, f'(3)=8. Find r'(1).

r'(x)=

Answer 1

Hello Abel,

For this problem, we want to move forward with differentiating r(x) and respecting the chain rule throughout since r(x) = f(g(h(x))) where 'f' is composed of g(x) and h(x).

Moving forward differentiating r(x):

We will get...

r'(x) = d/dx[f(g(h(x)))]

= f'(g(h(x)))*g'(h(x))*h'(x) where we are respecting each iteration of the inner most functions in sequential order of their compositions from outside to inside. (f -> g -> h)

Now, plugging in our value x = 1:

r'(1) = f'(g(h(1)))*g'(h(1))*h'(1), we now turn to what our given values are to evaluate and simply.

r'(1) = f'(g(4))*g'(4)*(3) since h(1) = 4 and h'(1) = 3 (Contrast the difference between this line and the written line above with 'h')

r'(1) = f'(3)*(3)*(3) since g(4) = 3 and g'(4) = 3

r'(1) = (8)*(3)*(3) since f'(3) = 8

r'(1) = 72 evaluating and simplifying.

Hope this helps!