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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.
Show Solution

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,

\[f'\left( x \right) = - \left( {2x{{\bf{e}}^x} + {x^2}{{\bf{e}}^x}} \right)\sin \left( {{x^2}{{\bf{e}}^x}} \right)\]