What is the derivative of f(x)= √X using definition of derivative?

By "definition of derivative" I'm assuming you mean one of the limit definitions, i.e. ƒ'(x) = lim(h→0) [(ƒ(x + h) − ƒ(x))/h]

Given:
ƒ(x) = √x

Apply the definition of derivative:
ƒ'(x) = lim(h→0) [(ƒ(x + h) − ƒ(x))/h]
= lim(h→0) [(√(x + h) − √x)/h]

Multiply by (√(x + h) + √x)/(√(x + h) + √x):
= lim(h→0) [(√(x + h) − √x)(√(x + h) + √x) / (h(√(x + h) + √x))]

FOIL out the numerator:
= lim(h→0) [(x + h − x) / (h(√(x + h) + √x))]
= lim(h→0) [h / (h(√(x + h) + √x))]

Cancel the h in the numerator and denominator:
= lim(h→0) [1 / (√(x + h) + √x)]

Let h go to 0:
= 1 / (√(x + 0) + √x)
= 1/(√x + √x)
= 1/(2√x)

Therefore, ƒ'(x) = 1/(2√x).

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f(x) = x^n

f'(x) = n*x ^ (n-1)

Apply the above theory to your problem...


f(x) = x ^ (1/2)

f'(x) = (1/2) * x ^ (1/2 - 1)

f'(x) = 1 / (2x^ (1/2) )