I'm trying to find the derivative of $f(x)=3\sqrt x$ at $25$. How would you go about this using the limit definition of a derivative? I'm currently stuck at $(3\sqrt{25+h}+15)/h$.
2026-05-06 05:07:04.1778044024
How to use limit definition to find derivative with a radical
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Note that we have
$$\begin{align} \lim_{h\to 0}\frac{f(25+h)-f(25)}{h}&=\lim_{h\to 0}\frac{3\sqrt{25+h}-3\sqrt{25}}{h}\\\\ &=\lim_{h\to 0}\left(\frac{3\sqrt{25+h}-3\sqrt{25}}{h}\,\times\frac{\sqrt{25+h}+\sqrt{25}}{\sqrt{25+h}+\sqrt{25}}\right)\\\\ &=\lim_{h\to 0}\frac{3}{\sqrt{25+h}+\sqrt{25}}\\\\ &=\frac{3}{10} \end{align}$$