Differentiation — Formula Sheet

Derivative at a point: lim_(x -> c) (f(x) - f(c)) / (x - c)

First principle: f'(x) = lim_(h -> 0) [f(x + h) - f(x)] / h

Right-hand and left-hand limits for differentiability: lim_(h -> 0)[(f(c + h) - f(c))/h] = lim_(h -> 0)[(f(c - h) - f(c))/(-h)]

Slope of secant PQ: tan θ = [f(a + h) - f(a)] / h

Instantaneous rate of change: dy/dx = lim_(Δx -> 0) Δy/Δx = lim_(Δx -> 0) [f(x + Δx) - f(x)] / Δx

d/dx(a) = 0

d/dx(x^n) = n x^(n - 1)

d/dx(sin x) = cos x

d/dx(cos x) = -sin x

d/dx(tan x) = sec^2 x

d/dx(f + g) = df/dx + dg/dx

d/dx(cf) = c df/dx

d/dx(fg) = f dg/dx + g df/dx

d/dx(f^n) = n f^(n - 1) df/dx

d/dx(tan(ax + b)) = a sec^2(ax + b)

d/dx(sin^2 x) = 2 sin x cos x

d/dx(cosec^2 x) = -2 cosec^2 x cot x

d/dx(sin(x^2)) = 2x cos(x^2)

Power rule

d/dx(x^n) = nx^(n-1).

A function f is differentiable at x = c if the limit lim_(x -> c) [f(x) - f(c)] / (x - c) exists finitely.

The derivative dy/dx at x = a represents the slope of the tangent to the curve y = f(x) at the point (a, f(a)).