derivative of 1)logx by first principle

let f(x) = log x

f(x + h) = log (x +h) {small increment }

by first principle

dy/dx = f(x + h) - f(x) / h as h tans to 0

= log (x +h) - log x / h h tans to 0

= log (x + h) /x whole divide by h h tans to 0 {using log m - log n = log (m)/n }

= log (1+h/x) / h h tans to 0

= log (1+h/x) / xh/x h tans to 0 {log (1+h/x) / h/x h tans to 0 = 1 ; using formula log (1 + x)/x x tans to 0 =1 }

therefore d ( log x) /dx = 1/x

hope might help you

= 1* 1/x = 1/x