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  2. Implicit function - Wikipedia

    en.wikipedia.org/wiki/Implicit_function

    An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments. [ 1 ] : 204–206 For example, the equation x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} of the unit circle defines y as an implicit function ...

  3. Implicit function theorem - Wikipedia

    en.wikipedia.org/wiki/Implicit_function_theorem

    The implicit derivative of y with respect to x, and that of x with respect to y, can be found by totally differentiating the implicit function + and equating to 0: + =, giving = and =. Application: change of coordinates

  4. General Leibniz rule - Wikipedia

    en.wikipedia.org/wiki/General_Leibniz_rule

    Second derivative; Implicit differentiation; ... If, for example, n = 2, the rule gives an expression for the second derivative of a product of two functions: ...

  5. Differentiation rules - Wikipedia

    en.wikipedia.org/wiki/Differentiation_rules

    Second derivative; Implicit differentiation; ... Some rules exist for computing the n-th derivative of functions, where n is a positive integer. These include:

  6. Implicit curve - Wikipedia

    en.wikipedia.org/wiki/Implicit_curve

    The implicit function theorem guarantees within a neighborhood of a point (,) the existence of a function such that (, ()) =. By the chain rule , the derivatives of function f {\displaystyle f} are

  7. Chain rule - Wikipedia

    en.wikipedia.org/wiki/Chain_rule

    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, ′ = ′ = (′) ′.

  8. Leibniz integral rule - Wikipedia

    en.wikipedia.org/wiki/Leibniz_integral_rule

    In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form () (,), where < (), < and the integrands are functions dependent on , the derivative of this integral is expressible as (() (,)) = (, ()) (, ()) + () (,) where the partial derivative indicates that inside the integral, only the ...

  9. Differential calculus - Wikipedia

    en.wikipedia.org/wiki/Differential_calculus

    (These two functions also happen to meet (−1, 0) and (1, 0), but this is not guaranteed by the implicit function theorem.) The implicit function theorem is closely related to the inverse function theorem, which states when a function looks like graphs of invertible functions pasted together.