Question
Differentiate $e^x \sin x+x^n \cos x$ with respect to $x$.
✓
Answer
Let, $\quad f(x)=e^x \sin x+x^n \cos x$
$\therefore \quad f^{\prime}(x)=\frac{d}{d x}\left\{e^x \sin x+x^n \cos x\right\}$
$=\frac{d}{d x}\left(e^x \sin x\right)+\frac{d}{d x}\left(x^n \cos x\right)$
$\begin{array}{l}=\sin x \frac{d}{d x} e^x+e^x \frac{d}{d x} \sin x+\cos x \frac{d}{d x} x^n+x^n \frac{d}{d x} \cos x \\ =\sin x \cdot e^x+e^x \cdot \cos x+\cos x \cdot n x^{n-1}+x^n \cdot(-\sin x) \\ =e^x(\sin x+\cos x)+x^{n-1}[n \cos x-x \sin x] .\end{array}$
$\therefore \quad f^{\prime}(x)=\frac{d}{d x}\left\{e^x \sin x+x^n \cos x\right\}$
$=\frac{d}{d x}\left(e^x \sin x\right)+\frac{d}{d x}\left(x^n \cos x\right)$
$\begin{array}{l}=\sin x \frac{d}{d x} e^x+e^x \frac{d}{d x} \sin x+\cos x \frac{d}{d x} x^n+x^n \frac{d}{d x} \cos x \\ =\sin x \cdot e^x+e^x \cdot \cos x+\cos x \cdot n x^{n-1}+x^n \cdot(-\sin x) \\ =e^x(\sin x+\cos x)+x^{n-1}[n \cos x-x \sin x] .\end{array}$
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