Differentiation Basics
Year 11 (IGCSE) 📊 Algebra Find derivatives using the power rule, gradients and turning points.
d/dx Introduction to Differentiation
Differentiation finds the gradient (rate of change) of a curve at any point.
Imagine a car journey. Differentiation finds the speed at any instant!
📐 The Power Rule
The most important rule for differentiating polynomials:
⚡ Power Rule
$$\frac{d}{dx}(x^n) = nx^{n-1}$$Examples:
$\frac{d}{dx}(x^3) = 3x^2$
$\frac{d}{dx}(x^5) = 5x^4$
$\frac{d}{dx}(4x^2) = 8x$
$\frac{d}{dx}(7) = 0$ (constants vanish)
$\frac{d}{dx}(x^3) = 3x^2$
$\frac{d}{dx}(x^5) = 5x^4$
$\frac{d}{dx}(4x^2) = 8x$
$\frac{d}{dx}(7) = 0$ (constants vanish)
📈 Gradient & Turning Points
The derivative $f'(x)$ gives the gradient at point $x$.
Turning Points
$$f'(x) = 0 \text{ at a turning point (maximum or minimum)}$$Example: $y = x^2 - 4x + 3$
$\frac{dy}{dx} = 2x - 4 = 0 \Rightarrow x = 2$
Minimum at $(2, -1)$
$\frac{dy}{dx} = 2x - 4 = 0 \Rightarrow x = 2$
Minimum at $(2, -1)$
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Interactive Demonstration — Differentiation Basics
d/dx Differentiation Calculator
Differentiate polynomials term by term using the power rule.