What Are Higher Derivatives?

Higher derivatives are derivatives obtained by differentiating a function repeatedly. The second derivative measures how the first derivative changes, while third and subsequent derivatives describe further rates of change.

How Higher Derivatives Work

For a function $y=f(x)$, the first derivative is

$$f'(x)=\frac{dy}{dx}.$$

Differentiating again gives the second derivative:

$$f''(x)=\frac{d^2y}{dx^2}.$$

Further differentiation produces

$$f'''(x)=\frac{d^3y}{dx^3}$$

and, generally, the nth derivative:

$$f^{(n)}(x)=\frac{d^ny}{dx^n}.$$

The notation $f^{(n)}(x)$ means “differentiate $f$ a total of $n$ times.” It does not mean raising $f(x)$ to the power .

The order tells you what is changing. For displacement $s(t)$, (s'(t)) is velocity, (s''(t)) is acceleration, and (s'''(t)) is the rate of change of acceleration, sometimes called jerk. Thus, higher derivatives extend the rate-of-change idea.

For example, let

$$f(x)=2x^4-3x^2+5.$$

Differentiating repeatedly:

$$f'(x)=8x^3-6x,$$ $$f''(x)=24x^2-6,$$ $$f'''(x)=48x,$$ $$f^{(4)}(x)=48.$$

Every derivative after the fourth is zero because the derivative of a constant is zero. This shows a useful polynomial pattern: each differentiation lowers the degree by one. A polynomial of degree $m$ has a constant $m$th derivative, and all derivatives of order greater than $m$ are zero.

The second derivative is particularly important. If $f''(x)>0$, the graph is concave-up; if $f''(x)<0$, it is . At a stationary point, indicates a local minimum and indicates a local maximum.

To classify a stationary point at $x=a$, first show $f'(a)=0$. If $f''(a)=0$, the second-derivative test is inconclusive; use another method, such as checking the sign of (f'(x)).

A common misconception is that $f''(x)=[f'(x)]^2$. This is incorrect: (f''(x)) is found by differentiating (f'(x)), not by squaring it.

IB Exam Technique

This is Mathematics: Analysis and Approaches HL content under AHL 5.12: First principles, higher derivatives, and it may be assessed on HL Papers 1, 2, or 3. For differentiate, show each required derivative clearly and use correct notation. Do not claim that $f''(x)=0$ alone proves a point of inflexion a change in concavity is also required.