Find the derivative of logx from first principle.

1.	Find the derivative of logx from first principle.
Answer» Let f(x) = log x ∴ \(f'(x)=\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}\) \(=\lim\limits_{h \to 0}\frac{log(x+h)-logx}{h}\) \(=\lim\limits_{h \to 0}\frac{log(\frac{x+h}{x})}{h}\) \(=\lim\limits_{h \to 0}\frac{log(1+\frac{h}{x})}{h}\) \(=\lim\limits_{h \to 0}\frac{log(1+\frac{h}{x})}{\frac{h}{x}}\times\frac{1}{x}\) \(=1\times\frac{1}{x}\) = \(\frac{1}{x}\)

Answer»

Let

f(x) = log x

∴ \(f'(x)=\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}\)

\(=\lim\limits_{h \to 0}\frac{log(x+h)-logx}{h}\)

\(=\lim\limits_{h \to 0}\frac{log(\frac{x+h}{x})}{h}\)

\(=\lim\limits_{h \to 0}\frac{log(1+\frac{h}{x})}{h}\)

\(=\lim\limits_{h \to 0}\frac{log(1+\frac{h}{x})}{\frac{h}{x}}\times\frac{1}{x}\)

\(=1\times\frac{1}{x}\)

= \(\frac{1}{x}\)

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