The first principle of differentiation, also known as the limit definition of a derivative, is a fundamental concept in calculus. It allows us to find the derivative of a function by calculating the limit of the difference quotient as the interval between two points approaches zero.
In this article, we will explore how to find the derivative of the function x√x by using the first principle of differentiation. We’ll break down the process into simple steps.
Answer: Derivative of x root(x) by First Principle is (3/2) x1/2
Now, let’s proceed with finding the derivative of x√x step by step:
First, we need to express the function x√x in its purest form. We’ve already done this by rewriting it as x(3/2).
The difference quotient, denoted as f'(x), is the expression we use to find the derivative. In this case, it is defined as:
f'(x) = lim (h -> 0) [(f(x + h) – f(x)) / h]
Substitute our function x^(3/2) into the difference quotient:
f'(x) = lim (h -> 0) [(x(3/2+h) – x(3/2)) / h]
Now, we simplify the expression within the limit as h approaches zero:
f'(x) = lim (h -> 0) [ x(3/2. (xh – 1)) / h]
To find the derivative, we calculate the limit of this expression as h approaches zero. This limit can be evaluated using L’Hôpital’s rule or by recognizing that it resembles the definition of the derivative of x^n.
After evaluating the limit, we arrive at the derivative of x√x:
f'(x) = (3/2) . x1/2