# 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 $$-\sin{x}$$.$\frac{1}{\cos{x}}+\frac{2\sin^{2}x}{\cos^{3}x}$Done1/cos(x)+(2*sin(x)^2)/cos(x)^3