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In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. [citation needed]The directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a direction ...
From this follows that the directional derivative is the inner product of its direction by the vector derivative. All needs to be observed is that the direction a {\displaystyle a} can be written a = ( a ⋅ e i ) e i {\displaystyle a=(a\cdot e^{i})e_{i}} , so that:
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
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.
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
Differentiation of integrals – Problem in mathematics; Differentiation under the integral sign – Differentiation under the integral sign formula; Hyperbolic functions – Collective name of 6 mathematical functions; Inverse hyperbolic functions – Mathematical functions
The derivatives of in the formula for must then be taken as weak derivatives. Another common function space is W g 1 , p ( Ω , R m ) {\displaystyle W_{g}^{1,p}(\Omega ,\mathbb {R} ^{m})} which is the affine sub space of W 1 , p ( Ω , R m ) {\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})} of functions whose trace is some fixed function g ...
A number of properties of the differential follow in a straightforward manner from the corresponding properties of the derivative, partial derivative, and total derivative. These include: [ 11 ] Linearity : For constants a and b and differentiable functions f and g , d ( a f + b g ) = a d f + b d g . {\displaystyle d(af+bg)=a\,df+b\,dg.}