Math 2.0

Real functions Concept

Implicit differentiation

Builds on Implicit function theoremChain rule in several variablesPartial derivative

Summary

Implicit differentiation finds the derivative of a function that an equation F(x,y)=0F(x, y) = 0 defines, without solving the equation for yy. One differentiates both sides of F(x,y(x))=0F(x, y(x)) = 0 with the chain rule and solves the result for y′(x)y'(x).

Remarks

If FF is of class C1C^1 and ∂F/∂y≠0\partial F / \partial y \ne 0 at the point, then y′(x)=−∂F/∂x∂F/∂yy'(x) = -\dfrac{\partial F / \partial x}{\partial F / \partial y}, evaluated at (x,y(x))(x, y(x)). The implicit function theorem guarantees that y(x)y(x) exists and is differentiable. For the circle x2+y2=1x^2 + y^2 = 1, differentiation gives 2x+2yy′=02 x + 2 y y' = 0. So y′=−x/yy' = -x / y at each point with y≠0y \ne 0, on both the upper and the lower arc. With several variables, F(x,y,z)=0F(x, y, z) = 0 gives ∂z/∂x=−Fx/Fz\partial z / \partial x = -F_x / F_z and ∂z/∂y=−Fy/Fz\partial z / \partial y = -F_y / F_z where Fz≠0F_z \ne 0. Here FxF_x, FyF_y, FzF_z are the partial derivatives of FF.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionChain rule in several variables, statement, no evidence levelChain rule in several…statementHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionJacobian determinant, definition, no evidence levelJacobian determinantdefinitionDerivative of the inverse function, statement, no evidence levelDerivative of the inv…statementC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementInverse function theorem, statement, no evidence levelInverse function theo…statementImplicit function theorem, statement, no evidence levelImplicit function the…statementImplicit differentiation, concept, no evidence levelImplicit differentiat…concept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Euclidean space object
  2. Function concept
  3. Function composition concept
  4. Fréchet derivative definition
  5. Function of several variables concept
  6. Limit of a function of several variables concept
  7. Partial derivative concept
  8. Differentiable function of several variables definition
  9. Partial derivative definition
  10. Differentiable on an open set definition
  11. Jacobian matrix definition
  12. Chain rule in several variables statement
  13. Higher partial derivatives definition
  14. Iterated derivative definition
  15. Jacobian determinant definition
  16. Derivative of the inverse function statement
  17. C^k function characterised by continuous derivatives statement
  18. Inverse function theorem statement
  19. Implicit function theorem statement
  20. Implicit differentiation concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Implicit_function, Implicit differentiation
  2. Web https://en.wikipedia.org/wiki/Implicit_function, Implicit function theorem
About this page
Tag0QKA345AW1
URIm2://0QKA345AW1
Aliasesimplicit-derivative
Kindconcept
WritingW0 stub
MSC26B10
UsesImplicit function theorem, Chain rule in several variables, Partial derivative

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