Question

Find from first principle the derivative of f(x) = log(x)

Answer 1

lim(h->0) log(x+h)-log(x)/h{base a }

log((x+h)/x) =>numerator =T

Using taylor series

−log(1−x)=x+x^2/2+x^3/ 3+…{but for using it we need natural log}

Using base change formula

log((x+h)/x) =ln((x+h)/x))/ln(a)

Now are limit looks like this,

lim(h->0)ln((x+h)/x))/h*ln(a)

ln((x+h)/x))=(x+h)/x-((x+h)/x)^2/2+((x+h)/x)^3/3-…

lim(h->0){(x+h)/x-((x+h)/x)^2/2+((x+h)/x)^3/3-…}/h*ln(a)

i think so it is helpful

Answer 2

Refer the attachment for complete answer