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Section 3.9 : Chain Rule

16. Differentiate \(f\left( x \right) = \ln \left( {\sin \left( x \right)} \right) - {\left( {{x^4} - 3x} \right)^{10}}\) .

Hint : Don’t get too locked into problems only requiring a single use of the Chain Rule. Sometimes separate terms will require different applications of the Chain Rule, or maybe only one of the terms will require the Chain Rule.
Show Solution

For this problem each term will require a separate application of the Chain Rule. The derivative is then,

\[f'\left( x \right) = \frac{{\cos \left( x \right)}}{{\sin \left( x \right)}} - 10\left( {4{x^3} - 3} \right){\left( {{x^4} - 3x} \right)^9} = \require{bbox} \bbox[2pt,border:1px solid black]{{\cot \left( x \right) - 10\left( {4{x^3} - 3} \right){{\left( {{x^4} - 3x} \right)}^9}}}\]