Question

Use the definition of the derivative to calculate the derivative of $f\left(x\right)=\sqrt{x}$ at $c=4$. At some point you will need to multiply numerator and denominator by the conjugate of $\sqrt{4+h}-2$, which is $\sqrt{4+h}+2$.

Short answer

The derivative is $\frac{1}{4}$.

Step-by-step solution

Step 1. Given Information. 

We are given a function,

$f\left(x\right)=\sqrt{x}$

Step 2. Finding the derivative. 

The derivative is given by,

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