differentiation using the first principle

Differentiation using first principles is a method of finding the derivative of a function by taking the limit of the difference quotient as the change in the input approaches zero. In other words, it is the process of finding the slope of a tangent line to a curve at a given point using the definition of the derivative.

To find the derivative of a function f(x) using first principles, we start by taking two points on the curve (x, f(x)) and (x + h, f(x + h)), where h is a small positive number. The slope of the secant line passing through these points is given by:

(f(x + h) – f(x)) / h

To find the slope of the tangent line at the point (x, f(x)), we take the limit of this expression as h approaches zero:

lim h->0 (f(x + h) – f(x)) / h

If this limit exists, it gives the derivative of the function f(x) at the point x, denoted as f'(x).

For example, let’s find the derivative of the function f(x) = x^2 using first principles:

lim h->0 (f(x + h) – f(x)) / h

= lim h->0 ((x + h)^2 – x^2) / h

= lim h->0 (x^2 + 2xh + h^2 – x^2) / h

= lim h->0 (2x + h)

= 2x

Therefore, the derivative of f(x) = x^2 is f'(x) = 2x.