The second derivative is f (x)= d dxf (x)= d2y dx2. It is not the square of the first derivative. **Derivative information** | **Meaning for the original graph** f >0 / f <0 | Increasing / decreasing f =0 | Stationary point; test the sign on either side f >0 | Gradient increasing; concave up f <0 | Gradient decreasing; concave down Example For f(x)=x3, f (x)=3x2 and f (x)=6x. The curve increases on both sides of zero; its concavity changes from down to up at zero. y=x3 y=3x2 The derivative graph touches the horizontal axis at the stationary inflection of x3. It does not cross there. Read derivatives from local gradients and signs; an asymptote of the original graph alone does not establish the limiting derivative.

The second derivative is \(\mathrm{f}^{\prime\prime}(x)=\dfrac{\mathrm d}{\mathrm dx}\mathrm{f}^{\prime}(x)=\dfrac{\mathrm d^2y}{\mathrm dx^2}\). It is not the square of the first derivative.

For \(\mathrm{f}(x)=x^3\), \(\mathrm{f}^{\prime}(x)=3x^2\) and \(\mathrm{f}^{\prime\prime}(x)=6x\). The curve increases on both sides of zero; its concavity changes from down to up at zero.

The derivative graph touches the horizontal axis at the stationary inflection of \(x^3\). It does not cross there. Read derivatives from local gradients and signs; an asymptote of the original graph alone does not establish the limiting derivative.

Applications of Differentiation

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Derivative information	Meaning for the original graph
\(\mathrm{f}^{\prime}>0\) / \(\mathrm{f}^{\prime}<0\)	Increasing / decreasing
\(\mathrm{f}^{\prime}=0\)	Stationary point; test the sign on either side
\(\mathrm{f}^{\prime\prime}>0\)	Gradient increasing; concave up
\(\mathrm{f}^{\prime\prime}<0\)	Gradient decreasing; concave down

Second Derivatives and Graphs

Second Derivatives and Graphs