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Question Number 162675 by Mathematification last updated on 31/Dec/21

$$y=\sqrt{x}$$$$Find\frac{dy}{dx}byfirstprinciple.$$

Answered by tounghoungko last updated on 31/Dec/21

$$\frac{dy}{dx}=\underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h}$$$$=\underset{h\to 0}{\lim }\frac{\sqrt{x+h}-\sqrt{x}}{h}=\underset{h\to 0}{\lim }\frac{h}{h(\sqrt{x+h}+\sqrt{x})}$$$$=\underset{h\to 0}{\lim }\frac{1}{\sqrt{x+h}+\sqrt{x}}=\frac{1}{2\sqrt{x}}$$

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