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Chris Fox's Engineering Section

Calculus: The Second Derivative

There may be points on a graph at which the curve reaches its highest or lowest y values, at least for a particular section. These points are called maxima and minima respectively; and at these points, dy/dx=0 (If dy/dx ≠ 0, it would mean the line is heading to or from a more extreme point, and the first point would not be a maximum or minimum). To find out which applies, differentiate the derivative to show the rate of change of gradient. The result is termed the second derivative, denoted by d²y/dx². Around a maximum, the gradient is first positive then negative, which means the rate of change is negative. Similarly, at a minimum d²y/dx²>0.

If d²y/dx²=0, the point is a point of inflexion, where the graph is heading in the same direction - up or down - on both sides but is temporarily level. To find out the direction of the graph, take the third derivative (The derivative of the second derivative). If d³y/dx³>0, it is going up.

Generally, a point is either a maximum or minimum if the lowest order derivative that does not equal 0 is even, and a point of inflexion if it is odd.

As with ordinary differentiation, second and higher order partial derivatives can be calculated. Here, however, second and higher order derivatives may use a different independent variable. For example, one can have (∂/∂y)(∂z/∂x), or fyx for short.

Second order partial derivatives can be used to check for maxima or minima where there is more than one independent variable. If fxxfyy-fxy²>0, the point is either a maximum or a minimum; use either fxx or fyy to find out which. If, on the other hand, fxxfyy-fxy²<0, the point is a saddle point: a minimum along one axis and a maximum along the other.

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