3.6 Calculating Higher-Order Derivatives

If f′ is differentiable, its derivative is the second derivative, f″. Differentiating again gives the third derivative, and so on.

Higher derivatives use more primes until that gets messy, then a number in parentheses: f⁽⁴⁾(x) is the fourth derivative. In Leibniz notation, the nth derivative of y is dⁿy/dxⁿ.

f′ is the rate of change of f, and f″ is the rate of change of f′. So f″ tells you whether the slopes are getting bigger or smaller.

In motion, if x(t) is position, then x′(t) is velocity and x″(t) is acceleration, the rate of change of velocity. On a graph, f″ > 0 means the slopes are increasing, so the graph bends upward (concave up), and f″ < 0 means it bends downward. Unit 5 builds on this.

You can estimate a second derivative from data too. If a table gives values of f′, then f″(a) ≈ (f′(b) − f′(c))/(b − c) using nearby inputs, exactly like estimating f′ from values of f in 2.3. With a velocity table, that's an estimate of acceleration.

For f(x) = x⁴ − 3x² + eˣ: f′(x) = 4x³ − 6x + eˣ, f″(x) = 12x² − 6 + eˣ and f‴(x) = 24x + eˣ. Simplify f′ before differentiating again; it makes the second step easier.

Some functions have patterns. The derivatives of sin x cycle every four steps: cos x, −sin x, −cos x, sin x. Each derivative of a polynomial lowers its degree by one, so a degree-5 polynomial's sixth derivative is 0.

For an implicit curve, d²y/dx² comes from differentiating dy/dx with respect to x. The result will contain dy/dx, because y is still a function of x. Replace dy/dx with the expression you already found, then simplify. Sometimes the original equation simplifies the answer further.

To evaluate at a point, you can also plug in numbers early: find dy/dx at the point first, then substitute x, y and dy/dx into the d²y/dx² expression. That avoids messy algebra.