# Differential Function, Product & Quotient Rules

The following table contains some rules and worked examples for differentiation.

## The Rules

 y = a dy/dx = 0Because 'a' is a constant, the slope of the equation (dy/dx) must be zero y = a.xn dy/dx = n.a.xn-1 y = a.xℓ + b.xm + c.xn dy/dx = ℓ.a.xℓ-1 + m.b.xm-1 + n.c.xn-1 y = u + v dy/dx = du/dx + dv/dx Function of a Function(algebraic) y = (axn - b)mu = axn - b > du/dx = a.n.xn-1y = um > dy/du = m.um-1dy/dx = dy/du . du/dx Function of a Function(trigonometric) y = Sin²(a + bx²) y = u² > dy/du = 2u u = Sin(v) > du/dv = Cos(v) v = a + bx² > dv/dx = 2bx dy/dx = dy/du . du/dv . dv/dx dy/dx = 2u . Cos(v) . 2.bx = 2.Sin(v) . Cos(a + bx²) . 2bx dy/dx = 2.Sin(a + bx²) . Cos(a + bx²) . 2bx dy/dx = 2bx . Sin(2(a + bx²)) Product Rule y = u.vdy/dx = u.dv/dx + v.du/dx Product Rule(one product) y = (3x² + 2x).( 6x - 2x³) u = 3x² + 2x v = 6x - 2x³ y = u.v dy/dx = u.dv/dx + v.du/dx dy/dx = d(3x² + 2x).(6x - 2x³)/dx + d(6x - 2x³).(3x² + 2x)/dx dy/dx = (6x + 2).(6x - 2x³) + (6 - 6x).(3x² + 2x) dy/dx = (36x² - 12x⁴ + 12x - 4x³) + (18x² + 12x - 18x³ - 12 x2) dy/dx = -12x⁴ - 4x³ - 18x³ + 36x² + 18x² - 12x2 + 12x + 12x dy/dx = -12x⁴ - 22x³ + 42x² +24x Product Rule(two products) y = (1 + x²).(2 - x²).(3 + x²) u = 1 + x² v = 2 - x² w = 3 + x² y = u.v.w dy/dx = u.v.dw/dx + u.w.dv/dx + v.w.du/dx dy = (1 + x²).(2 - x²).d(3 + x²)/dx + (1 + x²).(3 + x²).d(2 - x²)/dx + (2 - x²).(3 + x²).d(1 + x²)/dx dy/dx = (1 + x²).(2 - x²).(2x) + (1 + x²).(3 + x²).(2x) + (2 - x²).(3 + x²).(2x) dy/dx = 2x.(2 - x⁴ + 2x² - x²) + 2x.(3 + x⁴ + 3x² + x²) + 2x.(6 - x⁴ + 2x² - 3x²) dy/dx = (4x - 2x⁵ + 4x³ - 2x³) + (6x + 2x⁵ + 6x³ + 2x³) + (12x - 2x⁵ + 4x³ - 6x³) dy/dx = 4x + 6x + 12x + 4x³ - 2x³ - 6x³ + 2x³ + 4x³ + 6x³ - 2x⁵ + 2x⁵ - 2x⁵ dy/dx = 22x + 8x³ - 2x⁵ Quotient Rule y = u/v dy/dx = (v.du/dx - u.dv/dx) / v² y = 3x / (x - 1) u = 3x v = x - 1 dy/dx = [3(x-1) - 3x(1)] / (x² - 2x + 1) dy/dx = (3x - 3 - 3x) / (x² - 2x + 1) dy/dx = -3 / (x² - 2x + 1) Partial Differentiation 2.x³ - 3.xy + 5.x - 4.y² + 3 = 06.x²,    3.x.y € + 3.y,    5,    8.y.y € ,    0 6.x² - [3.x.y' + 3.y] + 5 - 8.y.y' + 0 = 0 6.x² - 3.x.y' - 3.y + 5 - 8.y.y' + 0 = 0 3.x.y' + 8.y.y' = 6.x² - 3.y + 5 y' (3.x + 8.y) = 6.x² - 3.y + 5 y' = (6.x² - 3.y + 5) / (3.x + 8.y)