Question

What are the methods of differentiation necessary to solve, and solve for the derivative using these methods. Thank you

Answer 1

Given:

`f(x)=e^x(3+2x^2)`

To find out the methods of differentiation necessary to solve:

Using the product rule, we get

`\begin{gathered} f^(\prime)(x)=(e^x)(d)/(dx)(3+2x^2)+(3+2x^2)(d)/(dx)(e^x) \\ f^(\prime)(x)=(e^x)(d)/(dx)(3)+(e^x)(d)/(dx)(2x^2)+(3+2x^2)(e^x) \end{gathered}`

Using power rule, we get,

`\begin{gathered} f^(\prime)(x)=0+4xe^x+(3+2x^2)(e^x) \\ =4xe^x+(3+2x^2)e^x \\ =e^x(2x^2+4x+3) \end{gathered}`

Hence, the answer is,

`f^(\prime)(x)=e^x(2x^2+4x+3)`