Differentiation from first principle

Definition of Derivative from First Principles:

The derivative of a function f(x) at a point x=a is defined as the limit of the difference quotient as the interval between two points approaches zero.

Derivative Definition: The derivative of f(x) at x=a is given by: f(a)=limh0f(a+h)f(a)h

Steps to Compute Derivatives Using First Principles:

  1. Find the Difference Quotient:

    • Start with the definition of the derivative: f(a+h)f(a)h.
  2. Simplify the Expression:

    • Expand f(a+h) and f(a).
    • Subtract f(a) from f(a+h).
  3. Factor and Simplify:

    • Factor out common terms.
    • Simplify the expression as much as possible.
  4. Take the Limit:

    • Let h approach zero (h0.
    • Evaluate the limit to find the derivative.

Example:

Let's find the derivative of a simple function f(x)=x2 from first principles at a point x=a.

  1. Difference Quotient: f(a)=limh0(a+h)2a2h

  2. Simplify: f(a)=limh0a2+2ah+h2a2h f(a)=limh02ah+h2h

  3. Factor and Simplify: f(a)=limh0h(2a+h)h f(a)=limh0(2a+h)=2a

  4. Take the Limit:

    • As h approaches zero, the derivative f(a)=2a.