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In calculus, a method called implicit differentiation makes use of the chain rule to differentiate implicitly defined functions. To differentiate an implicit function y ( x ) , defined by an equation R ( x , y ) = 0 , it is not generally possible to solve it explicitly for y and then differentiate.
The unit circle can be specified as the level curve f(x, y) = 1 of the function f(x, y) = x 2 + y 2.Around point A, y can be expressed as a function y(x).In this example this function can be written explicitly as () =; in many cases no such explicit expression exists, but one can still refer to the implicit function y(x).
The explicit midpoint method is sometimes also known as the modified Euler method, [1] the implicit method is the most simple collocation method, and, applied to Hamiltonian dynamics, a symplectic integrator. Note that the modified Euler method can refer to Heun's method, [2] for further clarity see List of Runge–Kutta methods.
Implicit differentiation; ... The difference rule ... These rules are given in many books, both on elementary and advanced calculus, in pure and applied mathematics. ...
Implicit differentiation; ... In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions.
Implicit differentiation; ... In calculus, the general Leibniz rule, [1] ... The rule can be proven by using the product rule and mathematical induction.
All extensions of calculus have a chain rule. In most of these, the formula remains the same, though the meaning of that formula may be vastly different. One generalization is to manifolds. In this situation, the chain rule represents the fact that the derivative of f ∘ g is the composite of the derivative of f and the derivative of g. This ...
Implicit differentiation; ... Calculus is the mathematical study of continuous change, ... and providing the product rule and chain rule, in their differential and ...
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