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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.
This is a list of Latin words with derivatives in English language. Ancient orthography did not distinguish between i and j or between u and v. [1] Many modern works distinguish u from v but not i from j. In this article, both distinctions are shown as they are helpful when tracing the origin of English words. See also Latin phonology and ...
Derivative (chemistry), a type of compound which is a product of the process of derivatization; Derivative (linguistics), the process of forming a new word on the basis of an existing word, e.g. happiness and unhappy from happy; Aeroderivative gas turbine, a mechanical drive gas turbine derived from an aero engine gas turbine
In chemistry, a derivative is a compound that is derived from a similar compound by a chemical reaction.. In the past, derivative also meant a compound that can be imagined to arise from another compound, if one atom or group of atoms is replaced with another atom or group of atoms, [1] but modern chemical language now uses the term structural analog for this meaning, thus eliminating ambiguity.
For this reason, the derivative is sometimes called the slope of the function f. [49]: 61–63 Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x 2 be the squaring function. The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is ...
for the first derivative, for the second derivative, for the third derivative, and for the nth derivative. When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken.
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.
If the derivative f vanishes at p, then f − f(p) belongs to the square I p 2 of this ideal. Hence the derivative of f at p may be captured by the equivalence class [f − f(p)] in the quotient space I p /I p 2, and the 1-jet of f (which encodes its value and its first derivative) is the equivalence class of f in the space of all functions ...