Problem of the Week

Updated at Jan 30, 2017 8:27 AM

How can we find the derivative of \(\sqrt{x}\tan{x}\)?

Below is the solution.



\[\frac{d}{dx} \sqrt{x}\tan{x}\]

1
Use Product Rule to find the derivative of \(\sqrt{x}\tan{x}\). The product rule states that \((fg)'=f'g+fg'\).
\[(\frac{d}{dx} \sqrt{x})\tan{x}+\sqrt{x}(\frac{d}{dx} \tan{x})\]

2
Since \(\sqrt{x}={x}^{\frac{1}{2}}\), using the Power Rule, \(\frac{d}{dx} {x}^{\frac{1}{2}}=\frac{1}{2}{x}^{-\frac{1}{2}}\)
\[\frac{\tan{x}}{2\sqrt{x}}+\sqrt{x}(\frac{d}{dx} \tan{x})\]

3
Use Trigonometric Differentiation: the derivative of \(\tan{x}\) is \(\sec^{2}x\).
\[\frac{\tan{x}}{2\sqrt{x}}+\sqrt{x}\sec^{2}x\]

Done
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