Differentiation by First Principles: Step-by-Step Guide
Differentiation by First Principles
Differentiation by first principles is an algebraic approach to finding the derivative (gradient function) of a function. The method calculates the gradient of a function as the limit of the difference quotient. The general formula is:
f'(x) = lim (h → 0) [f(x + h) - f(x)] / h
This formula computes the slope of the tangent line to the curve at any point x.
Derivation of the Differentiation Formula
- Consider the gradient between two points on the curve:
Gradient = (y₂ - y₁) / (x₂ - x₁)
- For the points (x, f(x)) and (x + h, f(x + h)), the gradient becomes:
Gradient = [f(x + h) - f(x)] / h
- As h approaches zero, the gradient approaches the derivative:
f'(x) = lim (h → 0) [f(x + h) - f(x)] / h
Steps to Differentiate by First Principles
- Find f(x + h): Replace x with x + h in the function f(x).
- Substitute f(x + h) and f(x) into the first principles formula.
- Simplify the numerator.
- Divide all terms by h.
- Evaluate the limit as h approaches zero.
Example 1: Differentiate f(x) = 2x + 5
- Find f(x + h):
f(x + h) = 2(x + h) + 5 = 2x + 2h + 5
- Substitute into the formula:
f'(x) = lim (h → 0) [(2x + 2h + 5) - (2x + 5)] / h
f'(x) = lim (h → 0) [2h] / h
- Simplify:
f'(x) = lim (h → 0) 2
- Evaluate the limit:
f'(x) = 2
Example 2: Differentiate f(x) = x² – x
- Find f(x + h):
f(x + h) = (x + h)² - (x + h) = x² + 2xh + h² - x - h
- Substitute into the formula:
f'(x) = lim (h → 0) [(x² + 2xh + h² - x - h) - (x² - x)] / h
f'(x) = lim (h → 0) [2xh + h² - h] / h
- Simplify:
f'(x) = lim (h → 0) (2x + h - 1)
- Evaluate the limit:
f'(x) = 2x - 1
Conclusion
Differentiation by first principles is a foundational technique in calculus, allowing us to find the derivative of a function from its basic definition. It provides insight into the behavior of functions and serves as a building block for more advanced differentiation techniques.
