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Section 3.3 : Differentiation Formulas

15. Find the tangent line to \(\displaystyle g\left( x \right) = \frac{{16}}{x} - 4\sqrt x \) at \(x = 4\).

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Hint : Recall the various interpretations of the derivative. One of them will help us do this problem.
Start Solution

Recall that one of the interpretations of the derivative is that it gives slope of the tangent line to the graph of the function.

So, we’ll need the derivative of the function. However before doing that we’ll need to do a little rewrite. Here is that work as well as the derivative.

\[g\left( x \right) = 16{x^{ - 1}} - 4{x^{\frac{1}{2}}}\hspace{0.5in} \Rightarrow \hspace{0.5in}\require{bbox} \bbox[2pt,border:1px solid black]{{g'\left( x \right) = - 16{x^{ - 2}} - 2{x^{ - \,\,\frac{1}{2}}} = - \frac{{16}}{{{x^2}}} - \frac{2}{{\sqrt x }}}}\]

Note that we rewrote the derivative back into rational expressions with roots to help with the evaluation.

Show Step 2

Next we need to evaluate the function and derivative at \(x = 4\).

\[g\left( 4 \right) = \frac{{16}}{4} - 4\sqrt 4 = - 4\hspace{0.5in}g'\left( 4 \right) = - \frac{{16}}{{{4^2}}} - \frac{2}{{\sqrt 4 }} = - 2\] Show Step 3

Now all that we need to do is write down the equation of the tangent line.

\[y = g\left( 4 \right) + g'\left( 4 \right)\left( {x - 4} \right) = - 4 - 2\left( {x - 4} \right)\hspace{0.25in} \to \hspace{0.25in}\,\require{bbox} \bbox[2pt,border:1px solid black]{{y = - 2x + 4}}\]