Paul's Online Notes
Paul's Online Notes
Home / Calculus I / Derivatives / Chain Rule
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

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,

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