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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,

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

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,


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.