Derivative of implicit functions.





Finding derivative of a explicit functions is easy using the differentiation techniques but
It is difficult to find the derivative of implicit  functions so we use the Implicit Differentiation technique to find the derivative of Implicit functions.

The Implicit Differentiation technique make use of the Chain Rule and the Sum Rule.

Example of Implicit Differentiation:

let us Differentiate y with respect to x in this Implicit function:

4x3-3y2=21

Differentiating both side of equation with respect to x , and using The Sum Rule:

\dfrac{d}{dx}\left(4x^3\right)-\dfrac{d}{dx}\left(3y^2\right) = \dfrac{d}{dx}\left(21\right)

Now using the The Chain Rule in the second term of the equation:

or,
12x^2-3 \times \dfrac{d}{dy}\left(y^2\right) \times \dfrac{d}{dx}\left(y\right) = o

or,
12x^2-6y \times \dfrac{d}{dx}\left(y\right) = o

SO,

\dfrac{d}{dx}\left(y\right)= \dfrac{12x^2}{6y}



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