3.6 Calculating Higher-Order Derivatives

What are higher-order derivatives?

Differentiating a function multiple times gives higher-order derivatives. The first derivative f'(x) gives slope/rate of change; the second derivative f''(x) is the derivative of f'(x), showing concavity and acceleration. Notation includes f''(x), y'', d²y/dx², or f⁽ⁿ⁾(x) for the nth derivative. Just keep differentiating the previous result.

Implicit differentiation for higher orders

To find d²y/dx² implicitly, differentiate the equation once to get dy/dx, then differentiate again with respect to x, substituting the expression for dy/dx you already found. This often introduces both x, y, and dy/dx terms — simplify by replacing dy/dx with its known expression before solving for d²y/dx².

Higher-order derivatives of parametric/vector functions

For parametric curves, d²y/dx² is NOT simply the derivative of dy/dx with respect to t. Instead, compute dy/dx = (dy/dt)/(dx/dt), then d²y/dx² = [d/dt(dy/dx)] / (dx/dt). Always divide by dx/dt again — a common error is skipping this final division.

Key terms

Second derivative notation

f''(x), y'', or d²y/dx² — the derivative of the first derivative.

nth derivative notation

f⁽ⁿ⁾(x) or dⁿy/dxⁿ, found by differentiating n times.

Implicit second derivative method

Differentiate the equation twice with respect to x, substituting the known dy/dx expression before solving for d²y/dx².

Parametric second derivative formula

d²y/dx² = [d/dt(dy/dx)] ÷ (dx/dt) — divide by dx/dt twice, not d²x/dt².

Physical meaning of second derivative

If position is s(t), s'(t) is velocity and s''(t) is acceleration.

Try an AP-style question

For f(x) = x⁴ − 3x², what is f''(x)?

1. 12x² − 6x
2. 4x³ − 6x
3. 24x
4. 12x² − 6