Problem of the Week

Updated at May 23, 2016 12:02 PM

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

Below is the solution.



\[\frac{d}{dx} \ln{x}\cos{x}\]

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

2
The derivative of \(\ln{x}\) is \(\frac{1}{x}\).
\[\frac{\cos{x}}{x}+\ln{x}(\frac{d}{dx} \cos{x})\]

3
Use Trigonometric Differentiation: the derivative of \(\cos{x}\) is \(-\sin{x}\).
\[\frac{\cos{x}}{x}-\ln{x}\sin{x}\]

Done
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