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The derivative is a fundamental tool of calculus that quantifies the sensitivity of 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.
The derivative of the function () = (()) is ′ = ′ (()) ′ (). In Leibniz's notation, this is written as: d d x h ( x ) = d d z f ( z ) | z = g ( x ) ⋅ d d x g ( x ) , {\displaystyle {\frac {d}{dx}}h(x)=\left.{\frac {d}{dz}}f(z)\right|_{z=g(x)}\cdot {\frac {d}{dx}}g(x),} often abridged to d h ( x ) d x = d f ( g ( x ) ) d g ( x ) ⋅ d g ...
The second derivative of a quadratic function is constant. In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f.
The quotient rule states that the derivative of h(x) is ′ = ′ () ′ (()). It is provable in many ways by using other derivative rules.
A function f of x, differentiated once in Lagrange's notation. One of the most common modern notations for differentiation is named after Joseph Louis Lagrange, even though it was actually invented by Euler and just popularized by the former. In Lagrange's notation, a prime mark denotes a derivative.
The natural base is a ubiquitous mathematical constant called Euler's number. To distinguish it, is called the exponential function or the natural exponential function: it is the unique real-valued function of a real variable whose derivative is itself and whose value at 0 is 1: for all , and.
In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function.
Differentiable functions can be locally approximated by linear functions. The function with for and is differentiable. However, this function is not continuously differentiable. A function is said to be continuously differentiable if the derivative exists and is itself a continuous function.