Differentiation

• dy/dx is the limit of the gradient of a chord as the chord shrinks to a point.
• d/dx(xⁿ) = nxⁿ⁻¹ for any rational n.
• Chain rule: dy/dx = dy/du × du/dx, so d/dx(ax + b)ⁿ = an(ax + b)ⁿ⁻¹.
• Tangent gradient = dy/dx; normal gradient = −1/(dy/dx).
• f is increasing where f′(x) > 0 and decreasing where f′(x) < 0.
• Connected rates of change: dA/dt = dA/dr × dr/dt.
• Stationary points where dy/dx = 0; d²y/dx² > 0 minimum, < 0 maximum.

From chord gradients to nxⁿ⁻¹

The gradient of the chord joining the points with x-coordinates 2 and 2 + h on y = x³ is [(2 + h)³ − 8]/h = 12 + 6h + h². As h → 0 this tends to 12, which is the gradient of the curve at x = 2. You need only this informal idea of a limit; then use the rules.

Before differentiating, rewrite roots and fractions as powers: √x = x^½, 2/x = 2x⁻¹. For a function of a function, use the chain rule.

Using the gradient

At the point (x₁, y₁), find m = dy/dx there; the tangent is y − y₁ = m(x − x₁) and the normal has gradient −1/m. For increasing or decreasing functions, solve the inequality f′(x) > 0 or f′(x) < 0.

For connected rates of change, link the rate you want to the rate you know with the chain rule, then substitute the given value at the end.

Where dy/dx = 0

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