Second Derivatives and Graphs

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.