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Section 3.2 : Interpretation of the Derivative

10. What is the equation of the tangent line to \(\displaystyle f\left( x \right) = \frac{5}{x}\) at \(\displaystyle x = \frac{1}{2}\)?

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We know that the derivative of a function gives us the slope of the tangent line and so we’ll first need the derivative of this function. We computed this derivative in Problem 8 from the previous section and so we won’t show the work here. If you need the practice you should go back and redo that problem before proceeding.

Note that we did use a different set of letters in the previous problem, but the work is identical. So, from our previous work (with a corresponding change of variables) we know that the derivative is,

\[f'\left( x \right) = - \frac{5}{{{x^2}}}\]

This tells us that the slope of the tangent line at \(x = \frac{1}{2}\) is then : \(m = f'\left( {\frac{1}{2}} \right) = - 20\). We also know that a point on the tangent line is : \(\left( {\frac{1}{2},f\left( {\frac{1}{2}} \right)} \right) = \left( {\frac{1}{2},10} \right)\).

The tangent line is then,

\[y = 10 - 20\left( {x - \frac{1}{2}} \right) = 20 - 20x\]