Differentiation

Derivatives (MM6.1, MM6.2)

• \(\dfrac{d}{dx}x^n = nx^{n-1}\) for any rational \(n\); constants differentiate to \(0\); sums term by term.
• Rewrite first: \(\dfrac{(3x - 2)^2}{x^2} = 9 - 12x^{-1} + 4x^{-2}\); \(\sqrt x = x^{1/2}\); \(\dfrac{1}{\sqrt x} = x^{-1/2}\).
• \(f'(a)\) is the gradient of the tangent at \(x = a\) and the rate of change there; velocity is \(\frac{ds}{dt}\).
• Second derivative \(f''(x)\): positive at a stationary point means a minimum, negative means a maximum, zero is inconclusive (check the sign change of \(f'\)).

Applications (MM6.3, MM8.5)

• Tangent at \((a, f(a))\): \(y - f(a) = f'(a)(x - a)\); normal: gradient \(-\frac{1}{f'(a)}\).
• Stationary points: solve \(f'(x) = 0\). Increasing where \(f'(x) > 0\), decreasing where \(f'(x) < 0\).
• A cubic \(ax^3 + bx^2 + cx + d\) has two turning points iff \(b^2 > 3ac\); it has three real roots iff the local max is above the axis and the local min below.
• Optimisation: express the quantity in one variable, differentiate, solve, confirm max/min, then evaluate.
• \(f'(x) = (x - 2)^2(x + 1)\): the squared factor does not change sign, so \(x = 2\) is a stationary point but not a turning point.