 # 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,

$$\begin{array}{l}x^{2}-5xy+3y^{2}=7\end{array}$$
$$\begin{array}{l}2x-\left(5x\frac{dy}{dx}+5y\right)+6y\frac{dy}{dx}=0\end{array}$$
$$\begin{array}{l}2x-5x\frac{dy}{dx}-5y+6y\frac{dy}{dx}=0\end{array}$$
$$\begin{array}{l}\frac{dy}{dx}\left(-5x+6y\right)=-2x+5y\end{array}$$
$$\begin{array}{l}\frac{dy}{dx}=\frac{-2x+5y}{-5x+6y}\end{array}$$
$$\begin{array}{l}\frac{dy}{dx}=\frac{2x-5y}{5x-6y}\end{array}$$