Differentiate w.r.t x:

Let $y=e^{(x \sin x+\cos x)}, z=x \sin x+\cos x, m=x$ and $w=\sin x$

Formula :

$\frac{\mathrm{d}\left(\mathrm{e}^{\mathrm{x}}\right)}{\mathrm{dx}}=\mathrm{e}^{\mathrm{x}}, \frac{\mathrm{d}(\sin \mathrm{x})}{\mathrm{dx}}=\cos \mathrm{x}$ and $\frac{\mathrm{d}(\cos \mathrm{x})}{\mathrm{dx}}=-\sin \mathrm{x}$

According to the product rule of differentiation

$\mathrm{dz} / \mathrm{dx}=\mathrm{w} \times \frac{\mathrm{dm}}{\mathrm{dx}}+\mathrm{m} \times \frac{\mathrm{dw}}{\mathrm{dx}}+\frac{\mathrm{d}(\cos \mathrm{x})}{\mathrm{dx}}$

$=[\sin x \times(1)]+[x \times(\cos x)]-\sin x$

$=x \cos x$

According to the chain rule of differentiation

$\mathrm{dy} / \mathrm{dx}=\frac{\mathrm{dy}}{\mathrm{dz}} \times \frac{\mathrm{dz}}{\mathrm{dx}}$

$=e^{(x \sin x+\cos x)} \times(x \cos x)$