Problem of the Week

Updated at Jun 27, 2016 8:23 AM

This week we have another calculus problem:

How can we solve for the derivative of \(\frac{{x}^{5}}{\cos{x}}\)?

Let's start!



\[\frac{d}{dx} \frac{{x}^{5}}{\cos{x}}\]

1
Use Quotient Rule to find the derivative of \(\frac{{x}^{5}}{\cos{x}}\). The quotient rule states that \((\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}\).
\[\frac{\cos{x}(\frac{d}{dx} {x}^{5})-{x}^{5}(\frac{d}{dx} \cos{x})}{\cos^{2}x}\]

2
Use Power Rule: \(\frac{d}{dx} {x}^{n}=n{x}^{n-1}\).
\[\frac{5{x}^{4}\cos{x}-{x}^{5}(\frac{d}{dx} \cos{x})}{\cos^{2}x}\]

3
Use Trigonometric Differentiation: the derivative of \(\cos{x}\) is \(-\sin{x}\).
\[\frac{5{x}^{4}\cos{x}+{x}^{5}\sin{x}}{\cos^{2}x}\]

Done
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