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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.
A function that has infinitely many derivatives is called infinitely differentiable or smooth. [34] Any polynomial function is infinitely differentiable; taking derivatives repeatedly will eventually result in a constant function, and all subsequent derivatives of that function are zero. [35] One application of higher-order derivatives is in ...
There are many alternatives to the classical calculus of Newton and Leibniz; for example, each of the infinitely many non-Newtonian calculi. [1] Occasionally an alternative calculus is more suited than the classical calculus for expressing a given scientific or mathematical idea.
The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if is a holomorphic function, real-valued on the real line, which can be evaluated at points in the complex plane near , then there are stable methods.
This is a list of limits for common functions such as elementary functions. In this article, the terms a , b and c are constants with respect to x . Limits for general functions
The q-derivative of a function is defined by the formula () = () (). For x nonzero, if f is a differentiable function of x then in the limit as q → 1 we obtain the ordinary derivative, thus the q-derivative may be viewed as its q-deformation.
The set of all velocities through a given point of space is known as the tangent space, and so df gives a linear function on the tangent space: a differential form. With this interpretation, the differential of f is known as the exterior derivative , and has broad application in differential geometry because the notion of velocities and the ...
Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.