\[\frac{d}{dx} x{e}^{x}\cos{x}\]
1
Use Product Rule to find the derivative of xexcos⁡xx{e}^{x}\cos{x}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
(ddxx)excos⁡x+x(ddxex)cos⁡x+xex(ddxcos⁡x)(\frac{d}{dx} x){e}^{x}\cos{x}+x(\frac{d}{dx} {e}^{x})\cos{x}+x{e}^{x}(\frac{d}{dx} \cos{x})

2
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
excos⁡x+x(ddxex)cos⁡x+xex(ddxcos⁡x){e}^{x}\cos{x}+x(\frac{d}{dx} {e}^{x})\cos{x}+x{e}^{x}(\frac{d}{dx} \cos{x})

3
The derivative of ex{e}^{x} is ex{e}^{x}.
excos⁡x+xexcos⁡x+xex(ddxcos⁡x){e}^{x}\cos{x}+x{e}^{x}\cos{x}+x{e}^{x}(\frac{d}{dx} \cos{x})

4
Use Trigonometric Differentiation: the derivative of cos⁡x\cos{x} is −sin⁡x-\sin{x}.
excos⁡x+xexcos⁡x−xexsin⁡x{e}^{x}\cos{x}+x{e}^{x}\cos{x}-x{e}^{x}\sin{x}

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