Question

Find the derivative of log_(a)x using first principle

Answer 1

`\displaystyle f'(x)=(1)/(x\ln(a))`

Explanation:

`\displaystyle f'(x)=\lim_(h\rightarrow0)(f(x+h)-f(x))/(h)\\\\f'(x)=\lim_(h\rightarrow0)(\log_a(x+h)-\log_a(x))/(h)\\\\f'(x)=\lim_(h\rightarrow0)(\log_a((x+h)/(x)))/(h)\\\\f'(x)=(1)/(h)\lim_(h\rightarrow0)\log_a\biggr(1+(h)/(x)\biggr)\\\\f'(x)=\lim_(h\rightarrow0)\log_a\biggr(1+(h)/(x)\biggr)^(1)/(h)\\\\f'(x)=\log_a(e^(1)/(x))\\\\f'(x)=(1)/(x)\log_a(e)\\\\f'(x)=(1)/(x)\biggr((\log(e))/(\log(a))\biggr)`

`\displaystyle f'(x)=(\log(e))/(x\log(a))\\\\f'(x)=(\ln(e))/(x\ln(a))\\\\f'(x)=(1)/(x\ln(a))`