Problem of the Week

Updated at Mar 13, 2017 3:23 PM

This week we have another calculus problem:

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

Let's start!



ddxx5sin⁡x\frac{d}{dx} \frac{{x}^{5}}{\sin{x}}

1
Use Quotient Rule to find the derivative of x5sin⁡x\frac{{x}^{5}}{\sin{x}}. The quotient rule states that (fg)′=f′g−fg′g2(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}.
sin⁡x(ddxx5)−x5(ddxsin⁡x)sin⁡2x\frac{\sin{x}(\frac{d}{dx} {x}^{5})-{x}^{5}(\frac{d}{dx} \sin{x})}{\sin^{2}x}

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

3
Use Trigonometric Differentiation: the derivative of sin⁡x\sin{x} is cos⁡x\cos{x}.
5x4sin⁡x−x5cos⁡xsin⁡2x\frac{5{x}^{4}\sin{x}-{x}^{5}\cos{x}}{\sin^{2}x}

Done
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