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

7. Differentiate \(f\left( t \right) = 5 + {{\bf{e}}^{4t + {t^{\,7}}}}\) .

Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule.
Show Solution

Note that we only need to use the Chain Rule on the second term as we can differentiate the first term without the Chain Rule.

Now, recall that for exponential functions outside function is the exponential function itself and the inside function is the exponent. The derivative is then,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f'\left( t \right) = \left( {4 + 7{t^6}} \right){{\bf{e}}^{4t + {t^{\,7}}}}}}\]