Question

Find the derivatives of the function:$f\left(x\right)=e^x\sin \left(x\right).$

Short answer

The derivative of $f\left(x\right)=e^x\sin \left(x\right)is{e}^x\{\sin \left(x\right)+\cos \left(x\right)\}.$

Step-by-step solution

Given information 

The given expression is$f\left(x\right)=e^x\sin \left(x\right).$

Simplification 

$$\begin{gathered} f\text{'}\left(x\right) \\ =\frac{d}{dx}\left\{e^x\sin \left(x\right)\right\} \\ =\sin \left(x\right)\frac{d}{dx}\left(e^x\right)+e^x\frac{d}{dx}\left\{\sin \left(x\right)\right\} \\ =e^x\sin \left(x\right)+e^x\cos \left(x\right) \\ =e^x\{\sin \left(x\right)+\cos \left(x\right)\} \\ Hence,f\text{'}\left(x\right)=e^x\{\sin \left(x\right)+\cos \left(x\right)\}. \end{gathered}$$