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Differential calculus

Differential calculus is concerned with rates of change. Geometrically, the rate of change of a function is defined as the ratio of the rise to the run, i.e. , in Cartesian coordinates where and are small increments in and . In calculus notation, the gradient of the curve is written

(Note, that since the d's are operators rather than variables, they do not cancel.) Here, and are infintessimaly small changes in and respectively, called differentials.
The diagram above shows a curve with equation . To find the gradient at P, i.e. , we need to let tend to 0. Hence, we can define the gradient of a function at a point by

The gradient of a function is also known as its derivative. Commonly used notation includes for the derivative with respect to and for the derivative with respect to . (Note, that derivate and differential are not the same thing. The derivative of with respect to is , but the differential of is .)

Using the definition of the derivative above, we can deduce the gradient function of any function for which it is defined. For example, for the function ,

So to find the gradient of at the point , we substitute the coordinate into the gradient function to obtain . The geometric significance of this is that at this point, for every unit change in , we need to increase by six units to remain on the tangent to the curve at this point.

Differentiation is a linear transformation. That is to say

where a and b are constants.

Subsections

Next: Standard results Up: Calculus - a brief Previous: What is calculus?
Alexander Frolkin 2001-03-13