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### Section 3-9 : Chain Rule

23. Differentiate $$z = \sqrt {5x + \tan \left( {4x} \right)}$$ .

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Hint : Sometimes the Chain Rule will need to be done multiple times before we finish taking the derivative.
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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 2

In 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.