Second-order derivatives

The essentials

If y = f(x), its derivative dy/dx = f′(x) is again a function of x, and differentiating once more gives the second-order derivative, written d²y/dx², f″(x) or y″. It measures how fast the slope itself is changing. In motion problems, if s is displacement, the first derivative is velocity and the second derivative is acceleration. In graphs, a positive second derivative means the curve bends upwards (concave up), and a negative one means it bends downwards.

Second derivatives are often used to verify that a given function satisfies a relation such as y″ + y = 0. You compute y′ and y″, substitute, and simplify.

Key points

• y = xⁿ gives y″ = n(n − 1)xⁿ⁻².
• y = sin x gives y″ = −sin x; y = cos x gives y″ = −cos x, so both satisfy y″ + y = 0.
• y = e^(kx) gives y″ = k²e^(kx), so y″ = k²y.
• y = log x gives y′ = 1/x and y″ = −1/x².
• Parametric case: d²y/dx² = [d/dt (dy/dx)] ÷ (dx/dt). It is NOT (d²y/dt²) ÷ (d²x/dt²).

Exam tips

• Simplify the first derivative fully before differentiating again; it saves time and errors.
• In "show that" questions, express the answer in terms of y and y′ by substituting back the original function.

Common mistakes

• In parametric questions, forgetting to divide by dx/dt a second time.
• Sign slips with trigonometric functions: the derivative of cos x is −sin x.