Problem of the Week

Updated at Oct 28, 2013 9:02 AM

How can we solve for the derivative of \(\frac{\sin{x}}{\cos^{2}x}\)?

Below is the solution.



\[\frac{d}{dx} \frac{\sin{x}}{\cos^{2}x}\]

1
Use Quotient Rule to find the derivative of \(\frac{\sin{x}}{\cos^{2}x}\). The quotient rule states that \((\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}\).
\[\frac{\cos^{2}x(\frac{d}{dx} \sin{x})-\sin{x}(\frac{d}{dx} \cos^{2}x)}{\cos^{4}x}\]

2
Use Trigonometric Differentiation: the derivative of \(\sin{x}\) is \(\cos{x}\).
\[\frac{\cos^{3}x-\sin{x}(\frac{d}{dx} \cos^{2}x)}{\cos^{4}x}\]

3
Use Chain Rule on \(\frac{d}{dx} \cos^{2}x\). Let \(u=\cos{x}\). Use Power Rule: \(\frac{d}{du} {u}^{n}=n{u}^{n-1}\).
\[\frac{\cos^{3}x-\sin{x}\times 2\cos{x}(\frac{d}{dx} \cos{x})}{\cos^{4}x}\]

4
Use Trigonometric Differentiation: the derivative of \(\cos{x}\) is \(-\sin{x}\).
\[\frac{1}{\cos{x}}+\frac{2\sin^{2}x}{\cos^{3}x}\]

Done
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