Given:
[tex]f(x)=e^x(3+2x^2)[/tex]
To find out the methods of differentiation necessary to solve:
Using the product rule, we get
[tex]\begin{gathered} f^{\prime}(x)=(e^x)\frac{d}{dx}(3+2x^2)+(3+2x^2)\frac{d}{dx}(e^x) \\ f^{\prime}(x)=(e^x)\frac{d}{dx}(3)+(e^x)\frac{d}{dx}(2x^2)+(3+2x^2)(e^x) \end{gathered}[/tex]
Using power rule, we get,
[tex]\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}[/tex]
Hence, the answer is,
[tex]f^{\prime}(x)=e^x(2x^2+4x+3)[/tex]
