Implicit Differentiation Formula

Implicit differentiation is the procedure of differentiating an implicit equation with respect to the desired variable x while treating the other variables as unspecified functions of x.

To differentiate an implicit function, any of the following methods is followed :

  • In the first method, the implicit equation is solved for y and it is expressed explicitly in terms of x and differentiation of y is carried. This method is found useful only when y is easily expressible in terms of x.
  • In the second method, y is thought of as a function of x, and both members of the implicit equation are differentiated w.r.t x. The resulting equation is solved to find the value of
    \(\begin{array}{l}\frac{dy}{dx}\end{array} \)
    .

Solved Examples

Question 1: Calculate the implicit derivative of

\(\begin{array}{l}x^{2}-5xy+3y^{2}=7\end{array} \)
?

Solution:

Given implicit function is,

\(\begin{array}{l}x^{2}-5xy+3y^{2}=7\end{array} \)

\(\begin{array}{l}2x-\left(5x\frac{dy}{dx}+5y\right)+6y\frac{dy}{dx}=0\end{array} \)

\(\begin{array}{l}2x-5x\frac{dy}{dx}-5y+6y\frac{dy}{dx}=0\end{array} \)

\(\begin{array}{l}\frac{dy}{dx}\left(-5x+6y\right)=-2x+5y\end{array} \)

\(\begin{array}{l}\frac{dy}{dx}=\frac{-2x+5y}{-5x+6y}\end{array} \)

\(\begin{array}{l}\frac{dy}{dx}=\frac{2x-5y}{5x-6y}\end{array} \)

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