Question

Hello! High school student in Calculus here. I need help solving the problem attached in the image. If someone could help break down the steps and explain how to solve using the product rule and/or quotient rule as pictured, I would greatly appreciate it!

Answer 1

Given data:

The given function is f(x)= e^(x) lnx.

The derivative of the given function using product rule is,

$\begin{gathered} f^{\prime}(x)=e^x\frac{d}{dx}(\ln x)+\ln x\frac{d}{dx}(e^x) \\ =e^x(\frac{1}{x})+\ln x(e^x)^{} \\ =e^x(\frac{1}{x}+\ln x) \end{gathered}$

The given function can be written as,

$\begin{gathered} f^{\prime}(x)=\frac{d}{dx}(\frac{\ln x}{e^{-x}}) \\ =\frac{e^{-x}\frac{d}{dx}(\ln x)-\ln x\frac{d}{dx}(e^{-x})}{(e^{-x})^2} \\ =\frac{e^{-x}(\frac{1}{x})+e^{-x}\ln x}{e^{-2x}} \\ =e^x(\frac{1}{x}+\ln x) \end{gathered}$

Thus, the derivative of the given function using product rule or quotient rule is e^x (1/x + lnx).