DIFFERENTIATION 12

Higher Derivatives

A function can be differentiated more than once. Each derivative describes how the previous one changes.

The main idea

\[\boxed{f''(x)=\frac{d^2y}{dx^2}}\]

The second derivative measures how the first derivative changes.

1. Example

If \(y=x^3\), then \(y'=3x^2\) and \(y''=6x\).

2. Notation

\[f''(x),\qquad y'',\qquad\frac{d^2y}{dx^2}.\]

3. Higher orders

The \(n\)-th derivative may be written \(f^{(n)}(x)\) or \(d^ny/dx^n\).

4. Physical meaning

If \(s(t)\) is position, \(ds/dt\) is velocity and \(d^2s/dt^2\) is acceleration.

5. Concavity

If \(f''(x)\gt0\), the slope is increasing and the graph is locally concave upward. If \(f''(x)\lt0\), the slope is decreasing and the graph is locally concave downward.

6. Inflection points

For \(f(x)=x^3\), \(f''(x)=6x\). For \(x\lt0\), \(f''(x)\lt0\); for \(x\gt0\), \(f''(x)\gt0\). The concavity therefore changes at \(x=0\).

7. Why higher derivatives matter

\[f\rightarrow f'\rightarrow f''\rightarrow f'''\rightarrow\cdots\]

They are important in motion, optimisation, differential equations and modelling.