Fundamental Algebraic Rules and Theorems of Differentiation

Evaluating the limit definition for every function is computationally tedious; standard operational differentiation rules are derived directly from first principles:

(1) Constant Rule: The derivative of any constant real number c is zero: d/dx [c] = 0 (since a horizontal line y = c has zero slope).

(2) Power Rule: For any real exponent n: d/dx [xⁿ] = n x^{n - 1}. Examples: d/dx [x] = 1; d/dx [x⁵] = 5x⁴; d/dx [1/x] = d/dx [x⁻¹] = -1 x⁻² = -1/x²; d/dx [√x] = d/dx [x^1/2] = (1/2) x^-1/2 = 1 / (2√x).

(3) Constant Multiple Rule: d/dx [c · f(x)] = c · f'(x).

(4) Sum and Difference Rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).

(5) Product Rule: The derivative of a product of two differentiable functions: d/dx [u(x) · v(x)] = u(x) · v'(x) + v(x) · u'(x) (First × derivative of second + Second × derivative of first).

(6) Quotient Rule: For rational functions where denominator v(x) ≠ 0: d/dx [u(x) / v(x)] = [v(x) · u'(x) - u(x) · v'(x)] / [v(x)]² (Denominator × derivative of numerator - Numerator × derivative of denominator, divided by denominator squared).