Derivative of a constant

The rule can be justified in various ways. The derivative is the slope of the tangent to the given function's graph, and the graph of a constant function is a horizontal line, whose slope is zero. Alternatively, one can use the limit definition of the derivative.

This leads to an important distinction between general and specific integrals. If the derivative of all constants were not the same, then there would be no ambiguity as to which constant the arbitrary constant of integration equalled. However, since the derivative of a constant is 0, then all constants have the same derivative, and so the arbitrary constant of integration is indeed arbitrary.

It also has the consequence that there is no polynomial function f such that f'(x) = x^-1. Generally, we would expect that if a function f existed such that f'(x) = x^k then that function would have degree k+1. However, in the x^-1 case, k+1=0, and the derivative of any function of degree 0 is the derivative of a constant which is 0, also of degree 0. So therefore that function f cannot be a polynomial function. In fact, the function f is the natural logarithm.