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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.
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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.