Problem of the Week

Updated at Jun 23, 2014 1:03 PM

This week we have another calculus problem:

How can we find the derivative of \(\frac{\csc{x}}{\ln{x}}\)?

Let's start!



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

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

2
Use Trigonometric Differentiation: the derivative of \(\csc{x}\) is \(-\csc{x}\cot{x}\).
\[\frac{-\ln{x}\csc{x}\cot{x}-\csc{x}(\frac{d}{dx} \ln{x})}{{\ln{x}}^{2}}\]

3
The derivative of \(\ln{x}\) is \(\frac{1}{x}\).
\[\frac{-\ln{x}\csc{x}\cot{x}-\frac{\csc{x}}{x}}{{\ln{x}}^{2}}\]

Done
Cymath foriOSAndroid