Problem of the Week

Updated at Nov 13, 2017 2:42 PM

How can we find the derivative of x5sin⁡x{x}^{5}\sin{x}?

Below is the solution.



ddxx5sin⁡x\frac{d}{dx} {x}^{5}\sin{x}

1
Use Product Rule to find the derivative of x5sin⁡x{x}^{5}\sin{x}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
(ddxx5)sin⁡x+x5(ddxsin⁡x)(\frac{d}{dx} {x}^{5})\sin{x}+{x}^{5}(\frac{d}{dx} \sin{x})

2
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
5x4sin⁡x+x5(ddxsin⁡x)5{x}^{4}\sin{x}+{x}^{5}(\frac{d}{dx} \sin{x})

3
Use Trigonometric Differentiation: the derivative of sin⁡x\sin{x} is cos⁡x\cos{x}.
5x4sin⁡x+x5cos⁡x5{x}^{4}\sin{x}+{x}^{5}\cos{x}

Done
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