What is the law of L'Hopital's in derivative?

L'Hopital's Rule is a mathematical theorem that provides a method for evaluating limits of indeterminate forms by taking the derivative of the numerator and denominator separately. The rule states that if the limit of a quotient of two functions is in an indeterminate form (such as 0/0 or ∞/∞), then the limit of the quotient is equal to the limit of the derivative of the numerator divided by the derivative of the denominator, provided that the limit of the derivatives exists. This rule is commonly used in calculus to simplify the evaluation of limits in certain cases.

Let f(x) and g(x) be functions such that f(a)=g(a)=0 or f(a)=∞ and g(a)=∞ or f(a) and g(a) are indeterminate forms of type 0/0 or ∞/∞. Then

limx→af(x)g(x)=limx→a[f′(x)g′(x)]......(i)
provided the limit on the right side exists or is infinite