Derivatives and Differentiation Flashcards

Grade 12 · Math · Flashcards

Master all essential derivative rules, formulas, and concepts needed for Grade 12 calculus.

Flashcards

Definition of the Derivative: f'(x) = lim(h→0) [f(x+h) - f(x)] / h; the instantaneous rate of change at x
Power Rule: d/dx[xⁿ] = nxⁿ⁻¹
Constant Rule: d/dx[c] = 0 for any constant c
Sum/Difference Rule: d/dx[f ± g] = f' ± g'
Product Rule: d/dx[f·g] = f'g + fg'
Quotient Rule: d/dx[f/g] = (f'g − fg') / g²
Chain Rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)
d/dx[eˣ]: eˣ
d/dx[aˣ]: aˣ · ln(a) for constant a > 0
d/dx[ln x]: 1/x
d/dx[tan x]: sec²x
Implicit Differentiation: Differentiate both sides of an equation with respect to x, treating y as a function of x; dy/dx terms are collected and solved
Local Minimum Test: If f'(c)=0 and f''(c) > 0, then x=c is a local minimum
Rolle's Theorem: If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c in (a,b) where f'(c)=0