Q. 6 - Calculus

Question

Use the definition of the derivative to calculate the derivative of $f\left(x\right)=x^{-1/2}$ at $c = 4$. As in the previous calculation, you will need to multiply numerator and denominator by a conjugate at some point.

Step-by-Step Solution

Answer

The derivative is $\frac{1}{16}$. 

Step 1. Given Information.

We are given a function, 

$f\left(x\right)=x^{-1/2}$

Step 2. Finding the derivative.

The derivative is given by, 

$$\begin{gathered} f^{\text{'}}\left(c\right)=\underset{h\to 0}{\lim }\frac{f(c+h)-f\left(c\right)}{h} \\ {f}^{\text{'}}\left(4\right)=\underset{h\to 0}{\lim }\frac{\frac{1}{\sqrt{4+h}}-\frac{1}{\sqrt{4}}}{h} \\ =\underset{h\to 0}{\lim }\frac{\left(\frac{1}{\sqrt{4+h}}-\frac{1}{2}\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{2-\sqrt{4+h}}{2h\left(\sqrt{4+h}\right)}\times \frac{2+\sqrt{4+h}}{2+\sqrt{4+h}} \\ =\underset{h\to 0}{\lim }\frac{4-4+h}{2h\left(\sqrt{4+h}\right)(2+\sqrt{4+h})} \\ =\underset{h\to 0}{\lim }\frac{1}{2\left(\sqrt{4+h}\right)(2+\sqrt{4+h})} \\ =\frac{1}{2\sqrt{4}(2+\sqrt{4})} \\ =\frac{1}{16} \end{gathered}$$