Differentiating from first principles.

 

Find the derivative of \[ f(x) = \sqrt x \] using the limit definition of the derivative.

Substitute the expression into the definition.

\[ \begin{align} f'(x) &= \lim_{h \to 0} \frac{f(x+h)- f(x)}{h} \\ &= \lim_{h \to 0} \frac{\sqrt{x+h}- \sqrt x}{h} \end{align} \ \]

Multiply numerator and denominator by (sqrt(x+h)+ sqrt(x))

\[ \begin{align} &= \lim_{h \to 0} \frac{(\sqrt{x+h}- \sqrt x)(\sqrt{x+h}+ \sqrt x)} {h(\sqrt{x+h} + \sqrt x)} \\ &= \lim_{h \to 0} \frac{(x + h - x)} {h(\sqrt{x+h} + \sqrt x)} \\ &= \lim_{h \to 0} \frac{h} {h(\sqrt{x+h} + \sqrt x)} \\ \end{align} \ \]

Cancel h and apply the definition of the limit.

\[ \begin{align} &= \frac{1} {2\sqrt{x}} \\ \end{align} \ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

 

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Now You Try:

Find the derivative of \[ f(x) = \frac{1}{x} \] using the limit definition of the derivative.