Please help me evaluate in detail the derivative of y =√x using the first principle method?

Answer 1

Ah, derivatives from first principles:

dy/dx = [h→0]lim (√(x+h) - √x)/h

Multiply both the numerator and denominator by (√(x+h)+√x):

[h→0]lim ((x+h) - x)/(h(√(x+h)+√x))

Simplify:

[h→0]lim (h)/(h(√(x+h)+√x))
[h→0]lim 1/(√(x+h)+√x)

This limit may now be evaluated by direct substitution:

1/(√(x+0)+√x)
1/(√x+√x)
1/(2√x)

Answer 2

y = √x
dy/dx =
lim (√x - √x0)/(x - x0) =
x→x0
lim (√x - √x0) (√x + √x0) / (x - x0) (√x + √x0) =
x→x0
lim (x - x0) / (x - x0) (√x + √x0) =
x→x0
lim 1 / (√x + √x0) = 1 / (2√x ) = (1/2)x^-(1/2)
x→x0