Question

If F(x) = f(g(x)), where f(−4) = 8, f '(−4) = 3, f '(−3) = 5, g(−3) = −4, and g'(−3) = 6, find F '(−3). F '(−3)

Answer 1

F'(-3) = 18

Explanation:

Let g(x) = u and apply the chain rule

`F(x)=f(g(x))=f(u)\\F'(x)=(df(u))/(du)`

`(du)/(dx)=g'(x)`

`(df(u))/(du)*(du)/(dx) = (df(u))/(dx)\\F'(x)= (df(u))/(du)*g'(x)\\F'(x)= f'(u)*g'(x)\\F'(x)= f'(g(x))*g'(x)`

We now have all of the necessary definite values to solve the expression for x= -3:

`F'(-3)= f'(g(-3))*g'(-3)\\F'(-3)= f'(-4)*6\\F'(-3)= 3*6\\F'(-3)= 18`

Finally, we have that F'(-3)= 18.