In calculus

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, the derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 of a constant function

Constant function

In mathematics, a constant function is a function whose values do not vary and thus are constant. For example the function f = 4 is constant since f maps any value to 4...

 is zero

0 (number)

0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...

 (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

Slope

In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....

 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

Formal proof

A formal proof or derivation is a finite sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system...

, from the definition of a derivative, is:


In Leibniz notation

Leibniz notation

In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent "infinitely small" increments of x and y, just as Δx and Δy represent finite increments of x and y...

, it is written as:

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

Mean value theorem

In calculus, the mean value theorem states, roughly, that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the...

. 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

Fundamental theorem of calculus

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

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