Question

Suppose
f(1) = 16 and f '(1) = 9
. Find the derivatives of the following functions

a. g(x) = sqrt(f(x))
g prime (1) =

b. h(x) = 1/(f(x))
h prime (1) =

Answer 1

This is just like taking the derivative of x, but remember that since we have another function of x, we have to include the chain rule.

a). \[g(x)=\sqrt{f(x)} \rightarrow g'(x)=\frac{ 1 }{ 2\sqrt{f(x)} }*f'(x)\]\[g'(1)=\frac{ f'(1) }{ 2\sqrt{f(1)} } \rightarrow g'(1)=\frac{ 9 }{ 2\sqrt{16} }\]

Answer 2

b). After rearranging the equation, we can use the power rule\[h(x)=(f(x))^{-1} \rightarrow h'(x)=-1(f(x))^{-2}*f'(x)\] \[h'(1)=\frac{ -f'(1) }{ (f(1))^2 }\]
Now just plug in the values that you were given

This could also be obtained from the quotient rule if we decided to keep it as 1/f(x).