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(d)/(dx)(\ln x)+\ln x(d)/(dx)(e^x) \\ =e^x((1)/(x))+\ln x(e^x)^{} \\ =e^x((1)/(x)+\ln x) \end{gathered}`

The given function can be written as,

`\begin{gathered} f^(\prime)(x)=(d)/(dx)((\ln x)/(e^(-x))) \\ =(e^(-x)(d)/(dx)(\ln x)-\ln x(d)/(dx)(e^(-x)))/((e^(-x))^2) \\ =(e^(-x)((1)/(x))+e^(-x)\ln x)/(e^(-2x)) \\ =e^x((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).