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 viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 3.9 : Chain Rule
23. Differentiate \(z = \sqrt {5x + \tan \left( {4x} \right)} \) .
Show All Steps Hide All Steps
Hint : Sometimes the Chain Rule will need to be done multiple times before we finish taking the derivative.
This problem will require multiple uses of the Chain Rule and so we’ll step though the derivative process to make each use clear.
Here is the first step of the derivative and we’ll need to use the Chain Rule in this step.
\[\begin{align*}z & = {\left( {5x + \tan \left( {4x} \right)} \right)^{\frac{1}{2}}}\\ & \frac{{dz}}{{dx}} = \frac{1}{2}{\left( {5x + \tan \left( {4x} \right)} \right)^{ - \,\,\frac{1}{2}}}\frac{d}{{dx}}\left( {5x + \tan \left( {4x} \right)} \right)\end{align*}\] Show Step 2In this step we can see that we’ll need to use the Chain Rule on the second term.
The derivative is then,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{dz}}{{dx}} = \frac{1}{2}{{\left( {5x + \tan \left( {4x} \right)} \right)}^{ - \,\,\frac{1}{2}}}\left( {5 + 4{{\sec }^2}\left( {4x} \right)} \right)}}\]In this step we were using the Chain Rule on the second term and so when multiplying by the derivative of the inside function we only multiply the second term by the derivative of the inside function and not both terms.