Using the Definition to Compute the Derivative - Exercise 1


Exercise 1. Find the derivative of

\begin{displaymath}f(x)=\sqrt{x}.\end{displaymath}

Answer. Let us see how we can simplify the difference quotient

\begin{displaymath}\frac{f(x+h)-f(x)}{h}=\frac{\sqrt{x+h}-\sqrt{x}}{h}.\end{displaymath}

Rationalizing the numerator leads to

\begin{displaymath}\frac{\sqrt{x+h}-\sqrt{x}}{h}=\frac{(\sqrt{x+h}-\sqrt{x})(\sq...
...x})}{h(\sqrt{x+h}+\sqrt{x})}
=\frac{h}{h(\sqrt{x+h}+\sqrt{x})}.\end{displaymath}

Consequently

\begin{displaymath}f^\prime(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to
0}\frac{1}{(\sqrt{x+h}+\sqrt{x})}=\frac{1}{2\sqrt{x}}.\end{displaymath}


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