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.

Consequences for antiderivatives

All constants have the same derivative; so that in taking an antiderivative there is the indeterminacy of adding any constant. The arbitrary constant of integration is indeed arbitrary, as the name suggests. This leads to an important distinction between general and specific integrals: according to the way the constant is left unfixed, or specified.

There is no polynomial function f such that f'(x) = x^-1; in fact, one such function f is the natural logarithm. A polynomial f exists such that f'(x) = x^k, for k = 0, 1, 2, ... . That function has degree k+1. In the x^-1 case, k+1=0, and this pattern breaks down: the derivative of any function of degree 0 is the derivative of a constant. Which is 0: the anti-derivative of x^-1 can't be a polynomial

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