The gradient of a *smooth* curve, , at a point is the gradient of the tangent to the curve at the point . Point is on the curve and is a neighbouring point whose value is displaced a small quantity, .

The idea behind differentiation is that as becomes very small, the gradient of tends towards the gradient of the curve. In the limit as becomes infinitesimally close to zero, the gradient becomes the gradient of the curve.

We write:

there is a fair bit of analytic work missing (higher education) to make these ideas sound.

We also write:

STANDARD RESULTS

Standard results can be proved for different functions.

If then

If , then we need to consider the *small angle approximation* that is if radians is very small (infinitesimal), then and , and compound trigonometry from which follows,

The differentiation process described above is *linear* and extends to more complicated functions. That is to say that if, where ,