derive e^(√(x)) from first principle

Question

how to solve this question "find the derivative of  e^(√(x)) from first principle method?

Answer 1

Let y=e^√x. Let x→x+h and y correspondingly → y+k.

ln(y)=√x, ln(y+k)=√(x+h)=√(x(1+h/x))=√x(1+h/x)^½≈√x(1+h/(2x)) when h is small.

ln(y+k)-ln(y)=ln(1+k/y)=√x(1+h/(2x))-√x=h√x/(2x)=h/(2√x).

When k is small, ln(1+k/y)≈k/y.

Therefore k/y=h/(2√x), k/h=y/(2√x)=e^√x/(2√x).

k/h≈dy/dx as h,k→0, so dy/dx=e^√x/(2√x).