CALCULUS AB
1. DIFFERENTIATION
Meaning
Differentiation means rate of change.
Basic Rule (Power Rule)
d/dx (xⁿ) = n xⁿ⁻¹
Examples
d/dx (x²) = 2x
d/dx (x³) = 3x²
d/dx (5x²) = 10x
Basic Derivatives
d/dx (1) = 0
d/dx (x) = 1
d/dx (sin x) = cos x
d/dx (cos x) = -sin x
d/dx (eˣ) = eˣ
Rules
Product Rule
(uv)’ = u v’ + v u’
Quotient Rule
(u/v)’ = (v u’ − u v’) / v²
, Chain Rule
d/dx f(g(x)) = f’(g(x)) × g’(x)
Example
y = (x² + 1)³
dy/dx = 3(x² + 1)² × 2x
= 6x(x² + 1)²
2. INTEGRATION
Meaning
Integration is the reverse of differentiation.
Basic Rule
∫xⁿ dx = xⁿ⁺¹ / (n+1) + C
Examples
∫x² dx = x³/3 + C
∫x³ dx = x⁴/4 + C
∫5x dx = (5x²)/2 + C
Basic Integrals
∫1 dx = x + C
∫1/x dx = ln|x| + C
∫eˣ dx = eˣ + C
∫sin x dx = -cos x + C
∫cos x dx = sin x + C
1. DIFFERENTIATION
Meaning
Differentiation means rate of change.
Basic Rule (Power Rule)
d/dx (xⁿ) = n xⁿ⁻¹
Examples
d/dx (x²) = 2x
d/dx (x³) = 3x²
d/dx (5x²) = 10x
Basic Derivatives
d/dx (1) = 0
d/dx (x) = 1
d/dx (sin x) = cos x
d/dx (cos x) = -sin x
d/dx (eˣ) = eˣ
Rules
Product Rule
(uv)’ = u v’ + v u’
Quotient Rule
(u/v)’ = (v u’ − u v’) / v²
, Chain Rule
d/dx f(g(x)) = f’(g(x)) × g’(x)
Example
y = (x² + 1)³
dy/dx = 3(x² + 1)² × 2x
= 6x(x² + 1)²
2. INTEGRATION
Meaning
Integration is the reverse of differentiation.
Basic Rule
∫xⁿ dx = xⁿ⁺¹ / (n+1) + C
Examples
∫x² dx = x³/3 + C
∫x³ dx = x⁴/4 + C
∫5x dx = (5x²)/2 + C
Basic Integrals
∫1 dx = x + C
∫1/x dx = ln|x| + C
∫eˣ dx = eˣ + C
∫sin x dx = -cos x + C
∫cos x dx = sin x + C
