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How do you find the derivative of #y =sqrt(x)# using the definition of derivative?

Answer 1

Luke Alvarez

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Answer 2

Isaiah Allen

To find the derivative of (y = \sqrt{x}) using the definition of derivative, we use the limit definition:

[f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}]

Substituting (f(x) = \sqrt{x}) into the formula, we have:

[\begin{split} f'(x) & = \lim_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h} \ & = \lim_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h} \times \frac{\sqrt{x + h} + \sqrt{x}}{\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})} \ & = \lim_{h \to 0} \frac{1}{\sqrt{x + h} + \sqrt{x}} \ & = \frac{1}{2\sqrt{x}} \end{split}]

Therefore, the derivative of (y = \sqrt{x}) using the definition of derivative is (y' = \frac{1}{2\sqrt{x}}).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.