Derivative of a constant

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.

Proof

A formal proof, from the definition of a derivative, is:

$f'(x) = \lim_{h \to 0} \frac {f(x+h)-f(x)}{h} = \lim_{h \to 0} \frac {c-c}{h} = \lim_{h \to 0}0 = 0.$

In Leibniz notation, it is written as:

$\frac {d}{dx}(c)=0.$

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 ƒ 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 ƒ is continuous on [a,b] and ƒ = g' for some function g, then

$\int_a^b f(x)\, dx = g(b) - g(a).$