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:
In Leibniz notation, it is written as:
Antiderivative of zero
A partial converse to this statement is the following:
This is not a consequence of the original statement, but follows from the mean value theorem. It can be generalized to the statement that
or
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
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