Derivatives of polynomials and trigonometric functions

For a polynomial aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, the derivative is naₙxⁿ⁻¹ + (n − 1)aₙ₋₁xⁿ⁻² + … + a₁: apply the power rule term by term.

Worked: d/dx (6x¹⁰⁰ − x⁵⁵ + x) = 600x⁹⁹ − 55x⁵⁴ + 1. For f(x) = 1 + x + x² + … + x⁵⁰, f′(1) = 1 + 2 + … + 50 = 1275.

Worked: (x + 1)/x = 1 + 1/x, so its derivative is −1/x².

Trigonometric derivatives, from first principles with sin h/h → 1: d/dx sin x = cos x and d/dx cos x = −sin x.

By the quotient rule, d/dx tan x = sec² x and d/dx cot x = −cosec² x. By the product rule, d/dx sin² x = 2 sin x cos x = sin 2x.

Worked: d/dx (2x + 3)/(x − 2) = −7/(x − 2)²; d/dx (x sin x) = x cos x + sin x; d/dx sin 2x = 2(cos² x − sin² x).

Calculus was invented independently by Newton (1642 to 1727) and Leibnitz (1646 to 1717) in the seventeenth century. Its limit-based foundations were made rigorous later by Cauchy, Lagrange and Weierstrass.