Question

Find the derivatives of the functions:$f\left(x\right)=e^{3x}\ln (x^2+1)$

Short answer

The required answer is$\frac{2x{e}^{3x}}{x^2+1}+3e^{3x}\ln (x^2+1)$.

Step-by-step solution

Step 1. Given information. 

The given function is:$f\left(x\right)=e^{3x}\ln (x^2+1)$

Step 2. Find the derivative of the given function.

$$\begin{gathered} f\left(x\right)=e^{3x}\ln (x^2+1) \\ f\text{'}\left(x\right)=\frac{d}{dx}\left(e^{3x}\ln (x^2+1)\right) \\ =e^{3x}\frac{d}{dx}\ln (x^2+1)+\ln (x^2+1)\frac{d}{dx}{e}^{3x} \\ =e^{3x}\frac{1}{x^2+1}\times 2x+\ln (x^2+1)3e^{3x} \\ =\frac{2x{e}^{3x}}{x^2+1}+3e^{3x}\ln (x^2+1) \end{gathered}$$