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Derivatives of Implicit Equations
  Derivatives of Implicit Equations

Derivatives of Implicit Equations

Functions are usually defined in terms of another variable explicitly. It is called y is a function of x and can be expressed as an explicit function, y=f(x).

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The explicit function can also be described as a relation between x and y implicitly.

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Sometime an implicit equation can be transformed into an explicit equation easily.

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But, sometime it is not an easy job to transform an implicit equation into an explicit equation.

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Derivatives of Implicit Equations

For a simple implicit equation the derivative of function y can be obtained by transforming into an explicit equation.

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Similarly the derivative of function y can also be obtained from an implicit equation directly.

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For any implicit equation, the derivative of function y can be derived from the first principal.

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Similarly for any implicit equation, the derivative of function y can be derived from the rules of differentiation.

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Since an implicit equation can be considered as a function g(x,y) composed of p(x), q(f(x)) and r(x,f(x)), therefore for any implicit equation, the derivative of function y can be derived from the rules of differentiation by differentiating both sides of the equation. In the above examples the RHS  is constant, the derivatives with respect to x is therefore equal to zero. The derivatives can then be obtained by solving the resulting equation. This is call mehod of implicit differentiation. And the above examples are assumed y is a differentiable function of x. 


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ID: 110900009 Last Updated: 9/17/2011 Revision: 0 Ref:

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References

  1. S. James, 1999, Calculus
  2. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science & Engineering
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