G: Differentiation flashcards

Topic 8: G: Differentiation

The two meanings of a derivative

Geometric: the gradient of the tangent line. Physical: the instantaneous rate of change.

Points of inflection

Where f″(x) = 0 and the concavity changes (f″ changes sign).

Differentiation from first principles

f′(x) = lim (h → 0) [f(x + h) − f(x)]/h. Required for small positive integer powers of x and for sin x and cos x.

Standard derivatives of xⁿ, e^(kx), aᵏˣ, ln x, sin kx, cos kx and tan kx

nxⁿ⁻¹ k e^(kx) k aᵏˣ ln a 1/x k cos kx −k sin kx k sec² kx

Product rule and quotient rule

y = uv: dy/dx = u dv/dx + v du/dx. y = u/v: dy/dx = (v du/dx − u dv/dx)/v².

Chain rule, with an example

dy/dx = (dy/du) × (du/dx). d/dx (3x² + 1)⁵ = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴.

Convex and concave

Convex: f″(x) > 0 (the gradient is increasing). Concave: f″(x) < 0 (the gradient is decreasing).

How do you find and classify stationary points?

Solve f′(x) = 0. Then f″(x) > 0: minimum; f″(x) < 0: maximum. If f″(x) = 0, check the sign of f′(x) either side.

Implicit and parametric differentiation (first derivative only)

Implicit: differentiate both sides with respect to x; each y term gains a factor dy/dx (d/dx(y²) = 2y dy/dx). Parametric: dy/dx = (dy/dt) ÷ (dx/dt).

Sketching a gradient function from a curve

Where the curve increases, the gradient function is positive; where it decreases, negative; at each turning point the gradient function crosses zero.