Question

Letf(x) = x ^ (4x) Use logarithmic differentiation to determine the derivative.

Answer 1

Let y = x^(4x). If we take the logarithm on both sides of the equation, we get the following:

`\begin{gathered} \ln y=\ln(x^(4x))=4x\ln x \\ \Rightarrow\ln y=4x\ln x \end{gathered}`

applying implicit derivation, we can find the derivative y':

`\begin{gathered} (\ln y)^(\prime)=(4x\ln x)^(\prime) \\ \Rightarrow(y^(\prime))/(y)=4\ln x+(4x)/(x) \\ \Rightarrow y^(\prime)=y(4(\ln x+1)) \\ \Rightarrow y^(\prime)=4x^(4x)(\ln x+1) \end{gathered}`

therefore, f'(x) = 4x^(4x)(lnx + 1).

Then, for f'(1), we can make x = 1 on the derivative to get the following:

`f^(\prime)(1)=4(1)^(4(1))(\ln(1)+1)=4(0+1)=4`

therefore f'(1) = 4