Derivative of a constant
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In calculus, the derivative of a constant function is zero. (A constant function is one that does not depend on the independent variable, such as f(x) = 7.)
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. The difference quotient <math>\frac{f(x+h)-f(x)}{h}<math> is zero for every h, and so therefore must be the limit of this quotient as h tends to zero, that is, f'(x).
Antiderivative of zero
A partial converse to this statement is the following:
- If a function has a derivative of zero on an interval, it must be constant on that interval.
This is not a consequence of the original statement, but follows from the Mean value theorem. It can be generalized to the statement that
- If two functions have the same derivative on an interval, they must differ by a constant,
or
- If g is an antiderivative of f on and interval, then all antiderivatives of f on that interval are of the form g(x)+C, where C is a constant.
From this follows a weak version of the second Fundamental theorem of calculus: if f is continuous on [a,b] and f = g' for some function g, then
- <math>\int_a^b f(x) dx = g(b) - g(a)<math>.