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  2. Derivative - Wikipedia

    en.wikipedia.org/wiki/Derivative

    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.

  3. Differential calculus - Wikipedia

    en.wikipedia.org/wiki/Differential_calculus

    velocity is the derivative (with respect to time) of an object's displacement (distance from the original position) acceleration is the derivative (with respect to time) of an object's velocity, that is, the second derivative (with respect to time) of an object's position. For example, if an object's position on a line is given by

  4. Calculus - Wikipedia

    en.wikipedia.org/wiki/Calculus

    The derivative f′(x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point. Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain ...

  5. Differential of a function - Wikipedia

    en.wikipedia.org/wiki/Differential_of_a_function

    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.}

  6. Product rule - Wikipedia

    en.wikipedia.org/wiki/Product_rule

    In calculus, the product rule (or Leibniz rule [1] or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions.For two functions, it may be stated in Lagrange's notation as () ′ = ′ + ′ or in Leibniz's notation as () = +.

  7. General Leibniz rule - Wikipedia

    en.wikipedia.org/wiki/General_Leibniz_rule

    The proof of the general Leibniz rule [2]: 68–69 proceeds by induction. Let and be -times differentiable functions.The base case when = claims that: ′ = ′ + ′, which is the usual product rule and is known to be true.

  8. Differentiation rules - Wikipedia

    en.wikipedia.org/wiki/Differentiation_rules

    The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ⁡ ( y , x ) . {\displaystyle \arctan(y,x).}

  9. Differential (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Differential_(mathematics)

    This can be motivated by the algebro-geometric point of view on the derivative of a function f from R to R at a point p. For this, note first that f − f(p) belongs to the ideal I p of functions on R which vanish at p. If the derivative f vanishes at p, then f − f(p) belongs to the square I p 2 of this ideal.