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Section 3.9 : Chain Rule
22. Differentiate \(f\left( x \right) = \cos \left( {{x^2}{{\bf{e}}^x}} \right)\) .
Hint : Don’t forget the Product and Quotient Rule. Sometimes, in the process of using the Chain Rule, you’ll also need the Product and/or Quotient Rule.
For this problem we’ll start off using the Chain Rule, however when we differentiate the inside function we’ll need to do the Product Rule.
The derivative is then,
\[f'\left( x \right) = - \left( {2x{{\bf{e}}^x} + {x^2}{{\bf{e}}^x}} \right)\sin \left( {{x^2}{{\bf{e}}^x}} \right)\]