Question

Find the derivative of loga x , using first principle rule

Answer 1

Answer:

$\displaystyle \frac{d}{dx} \log_a x =\lim_{h\rightarrow 0}\frac1h\left(\log_a(x+h)-\log_a x\right)\\{}\quad=\lim_{h\rightarrow0}\frac1h\log_a\frac{x+h}{x}\\{}\quad=\lim_{h\rightarrow0}\log_a\left(1 + \frac{h}{x}\right)^{1/h}\\{}\quad=\log_a\lim_{h\rightarrow0}\left(1 + \frac{h}{x}\right)^{1/h}\\{}\quad=\log_a e^{1/x}\\{}\quad=\frac1x\log_a e$

This uses the fact that

$\displaystyle e^x = \lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^n = \lim_{h\rightarrow0}(1+hx)^{1/h}$