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The derivative of ′ is the second derivative, denoted as ″ , and the derivative of ″ is the third derivative, denoted as ‴ . By continuing this process, if it exists, the n {\displaystyle n} th derivative is the derivative of the ( n − 1 ) {\displaystyle (n-1)} th derivative or the derivative of order ...
The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and its angle is zero. In other words, the value of the constant function, y, will not change as the value of x increases or decreases.
It is particularly common when the equation y = f(x) is regarded as a functional relationship between dependent and independent variables y and x. Leibniz's notation makes this relationship explicit by writing the derivative as: [1].
the partial differential of y with respect to any one of the variables x 1 is the principal part of the change in y resulting from a change dx 1 in that one variable. The partial differential is therefore involving the partial derivative of y with respect to x 1.
If f is a differentiable function on ℝ (or an open interval) and x is a local maximum or a local minimum of f, then the derivative of f at x is zero. Points where f'(x) = 0 are called critical points or stationary points (and the value of f at x is called a critical value).
The most common differential operator is the action of taking the derivative. Common notations for taking the first derivative with respect to a variable x include: , , , and . When taking higher, nth order derivatives, the operator may be written:
By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function x ↦ x 2 sin ( 1 / x ) {\displaystyle x\mapsto x^{2}\sin(1/x)} is a Darboux function even though it is not continuous at one point.
In calculus, the reciprocal rule gives the derivative of the reciprocal of a function f in terms of the derivative of ... ≠ 0 then g(x) = 1/f(x) ...