**Summary**

- An implicit equation is an equation which is not in the form , it consists of two variable
*x*and*y*which cannot be separated. - Implicit Functions are differentiated by using ”chain rule” in combination with the ”product and quotient rule”.
- When we differentiate
*y*we write with the derivative i.e

- To find the derivative of the product of
*x*and*y*i.e we use the product rule.

Previously, we have differentiated function where *y* was explicitly separated from *x*. Some examples of such functions are:

etc.

All these functions have *y* separated from *x* and we can then easily find derivative of *y* with respect to *x* .

On the other hand, consider an equation , we can see that we cannot separate *y* from *x* and we cannot write it in the form , and therefore we cannot also find directly differentiating the above equation. Such an equation is called and implicit function.

So to differentiate such an equation is known as ”Implicit Differentiation”.

Implicit Functions are differentiated by using ”chain rule” in combination with the product and quotient rule.

When we differentiate* y* we write with the derivative i.e

Let’s move on to solve an example so we can apply these rules to differentiate an implicit function.

#### Example #1

Q. Differentiate

*Solution:*

We see that the above equation is implicit hence we differentiate each term with respect to *x* like:

Use the product rule when differentiating *(xy)*:

Make the subject of the equation:

**Ans**

#### Example #2

Q. Find the equation of the tangent to the curve at the point (3, 3)

*Solution:*

Move the terms with together:

Make the subject of the equation:

Simplify the expression:

We can find the value of the gradient of the tangent by substituting *x* and *y *with the given point (3, 3):

So the tangent line is the line with slope −1 through the point (3, 3).

Using the formula for equation of a line, we find the equation of the tangent to the curve at (3, 3):

and

** Ans**

##### Reference