Question

If f(x) = 2^(8x^3+1), then find the exact value of f'(0.5)

Answer 1

Umm, do you know how to find the derivative?

Answer 2

the derivative of \(b^x\) is \(b^x\ln(b)\)
then chain rule

Answer 3

One way to look at it is: \[
2^{f(x)} = \left(e^{\ln(2)}\right)^{f(x)} = e^{\ln(2)f(x)}
\]Chain rule says: \[
\left(e^{\ln(2)f(x)}\right)' = e^{\ln(2)f(x)}(\ln(2)f(x))' =2^{f(x)}\ln(2)f'(x)
\]