# Implicit Differentiation Formula

The implicit differentiation formula is the method of differentiating an implicit equation with respect to the desired variable 'x' while treating the other variables as constant. To differentiate an implicit function, any of the two methods can be used. For the first method, the implicit equation is solved for variable 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 and for the second method, y is taken as a function of x, and both variables of the implicit equation are differentiated w.r.t x. The resulting equation is solved to find the value of \(\frac{dy}{dx}\). Let us learn more about the implicit differentiation formula in the upcoming sections.

## What Is the Implicit Differentiation Formula?

Implicit differentiation is the procedure of differentiating an implicit equation with respect to the desired variable x. But there is no particular formula to differentiate the implicit functions but we need to use chain rule while differentiating an implicit function.

\(\dfrac{dy}{dx}=\frac{dy}{du}.\dfrac{du}{dx}\)

Let us see how to use the implicit differentiation formula in the following solved examples section.

## Examples Using Implicit Differentiation Formula

**Example 1:** Differentiate the following function w.r.t x. using implicit differentiation formula.

y + cosy = sinx

**Solution:**

To find: \(\dfrac{dy}{dx}\)

y + cosy = sinx

Differentiating wrt x,

\(\dfrac{d}{dx}(y) + \dfrac{d}{dx}(cosy) = \dfrac{d}{dx}(sinx)\)

Using the implicit differentiation formula,and apply the chain rule for middle term.

\(\dfrac{dy}{dx} – siny \dfrac{d}{dx}(y)= cos x \)

\(\dfrac{dy}{dx} – siny \dfrac{dy}{dx} = cos x\)

\(\dfrac{dy}{dx}( 1-siny) = cos x\)

\(\dfrac{dy}{dx} = \dfrac{cosx}{1-siny}\) ( where y ≠ \(\dfrac{\pi}{2} ,\dfrac{3\pi}{2}\) and so on or sin y ≠1)

**Answer: **The differentiation of y + cosy = sinx is \(\dfrac{cosx}{1-siny}\).

**Example 2:** Find \(\dfrac{dy}{dx}\) of x^{2} + y^{2} = 9.

**Solution:**

Differentiating wrt x

\(\dfrac{d}{dx} (x^2) + \dfrac{d}{dx}(y^2) = \dfrac{d}{dx}(9)\)

Using the implicit differentiation formula,and apply the chain rule for y^{2}

2x + 2y \(\dfrac{d}{dx}\)(y) = 0

2x + 2y \(\dfrac{dy}{dx}\) =0

2y \(\dfrac{dy}{dx}\) = -2x

\(\dfrac{dy}{dx} = -\dfrac{2x}{2y}\)

\(\dfrac{dy}{dx} = \dfrac{-x}{y}\) ( where y≠ 0)

**Answer: **The \(\dfrac{dy}{dx}\) of x^{2} + y^{2} = 9 is \(\dfrac{-x}{y}\).