d2ydx2= ddx(dydx) The first derivative gives the gradient; the second gives the rate at which that gradient changes. Method Find the first derivative as a function; differentiate that expression again; only then evaluate at a given input. Example y=x3+2x, y =3x2+2, y =6x Exam check The second derivative is not the square of the first derivative. Substituting a point too soon turns the first derivative into a constant and loses its variation.

\(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}=\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)\)

The first derivative gives the gradient; the second gives the rate at which that gradient changes.

Find the first derivative as a function; differentiate that expression again; only then evaluate at a given input.

\(y=x^3+2x,\quad y^{\prime}=3x^2+2,\quad y^{\prime\prime}=6x\)

The second derivative is not the square of the first derivative. Substituting a point too soon turns the first derivative into a constant and loses its variation.

Second derivatives

Second derivatives