Using the Definition to Compute the Derivative - Exercise 1

Exercise 1. Find the derivative of

$$f(x)=\sqrt{x}.$$

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

$$\frac{f(x+h)-f(x)}{h}=\frac{\sqrt{x+h}-\sqrt{x}}{h}.$$

Rationalizing the numerator leads to

$$\frac{\sqrt{x+h}-\sqrt{x}}{h}=\frac{(\sqrt{x+h}-\sqrt{x})(\sq... ...x})}{h(\sqrt{x+h}+\sqrt{x})} =\frac{h}{h(\sqrt{x+h}+\sqrt{x})}.$$

Consequently

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

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