Paul's Online Notes
Home / Calculus I / Derivatives / Chain Rule
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 3-9 : Chain Rule

12. Differentiate $$V\left( x \right) = \ln \left( {\sin \left( x \right) - \cot \left( x \right)} \right)$$ .

Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule.
Show Solution

For this problem the outside function is (hopefully) clearly the logarithm and the inside function is the stuff inside of the logarithm. The derivative is then,

$V\left( x \right) = \frac{1}{{\sin \left( x \right) - \cot \left( x \right)}}\left( {\cos \left( x \right) + {{\csc }^2}\left( x \right)} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{\cos \left( x \right) + {{\csc }^2}\left( x \right)}}{{\sin \left( x \right) - \cot \left( x \right)}}}}$

With logarithm problems remember that after differentiating the logarithm (i.e. the outside function) you need to substitute the inside function into the derivative. So, instead of getting just,

$\frac{1}{x}$

we get the following (i.e. we plugged the inside function into the derivative),

$\frac{1}{{\sin \left( x \right) - \cot \left( x \right)}}$

Then, we can’t forget of course to multiply by the derivative of the inside function.