Find the derivative Find the derivative of y sqrt x from th

Find the derivative:

Find the derivative of y = sqrt x, from the first principles.

Solution

We\'ll put f(x) = y = sqrt x

f\'(x) = lim [f(x) - f(0)]/(x-0), for x->0

f\'(x) = lim (sqrtx - sqrt0)/x

We\'ll substitute x by 0:

lim (sqrtx - sqrt0)/x = (sqrt0 - sqrt0)/0 = 0/0

Since we have an indetermination case, 0/0, we\'ll apply L\'Hospital rule:

lim f/g = lim f\'/g\'

lim sqrtx/x = lim (sqrt x)\'/x\'

lim (sqrt x)\'/x\' = lim (1/2sqrt x)/1

We\'ll substitute x by 0 and we\'ll get:

lim (1/2sqrt x)/1 = (1/2sqrt 0)

f\'(x) = 1/2 sqrt x

f\'(0) = 1/2