First of all, "second derivative", d²y/dx², is what you get when you differentiate the first derivative (dy/dx).

The second derivative can be used as an easier way of determining the stationary points of a curve.

A stationary point on a curve can be a maximum point, a minimum point or a point of inflection. Those occur when dy/dx = 0. Once you have established where there is a stationary point, the type of stationary point (maximum, minimum or point of inflection) can be determined using the second derivative.

Thus,

If d²y/dx²(second derivative of y in terms of x) is positive, then it is a minimum point

If d²y/dx²is negative, then it is a maximum point

If d²y/dx²is zero, then it could be a maximum, minimum or point of inflection.

If d²y/dx²is 0, you must test the values of dy/dx (first derivative) either side of the stationary point, as before in the stationary points section.

If dy/dx is possitive before and negative after the stationary point then the last is a maximum.

If dy/dx is negative before and possitive after the stationary point then the last is a minimum.

Differentiate 3x^2 with respect to x

Related Maths A Level answers

Solve the simultaneous equations: x^2 + y^2 = 10 and x + 2y = 5

A curve with equation y = f(x) passes through the point (4,25). Given that f'(x) = (3/8)*x^2 - 10x^(-1/2) + 1, find f(x).

What is the second derivative used for?

Find all solutions of the equation in the interval [0, 2π]. 5 cos^3 x = 5 cos x

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