Find the Derivative of the Function Using the First Principle of Derivative

To find the derivative of a function using the first principle of derivatives, we start with the definition of the derivative:

Given a function f(x), the derivative f'(x) is defined as:

f'(x) = lim (h → 0) [(f(x + h) – f(x)) / h]

This limit represents the slope of the tangent to the function at the point x.

Let’s go through the steps with a specific function, say f(x) = x^2.

1. First, calculate f(x + h):
• f(x + h) = (x + h)^2 = x^2 + 2xh + h^2
2. Next, find f(x + h) – f(x):
• f(x + h) – f(x) = (x^2 + 2xh + h^2) – x^2 = 2xh + h^2
3. Now, divide by h:
• ((f(x + h) – f(x)) / h) = (2xh + h^2) / h = 2x + h
4. Finally, take the limit as h approaches 0:
• lim (h → 0) (2x + h) = 2x

Thus, the derivative of the function f(x) = x^2 using the first principle of derivatives is:

f'(x) = 2x