# Implicit Differentiation

**Introduction**

We have worked with functions where y can be expressed explicitly in terms of x.

For example, lets take a look at the following functions

- 1.

*x + y = 2*which can be written as

*y = 2 x.*

- 2. y
= x

^{2}

- 3.

*y = tan x + ln x*

All the above functions
are explicit functions.

But sometimes it is not as simple to write y in terms of x.

For example,

- 1.

*x*

^{2}+ y^{3}+ xy = 2- 2.

*tan(x + y) tan(x y) = 1*

Now, it is possible to find in case of implicit functions as well.

**Example 1:**

Find
when
*x ^{2} + y^{3} + xy = 16*

**Solution:**

Given:
*x ^{2} + y^{3} + xy = 16*

Differentiating both sides with respect to x we get,

*2x + 3y*

^{2}+ x + y(1) = 0**Note:**

- 1.
We differentiated y

^{3}by using the power rule and attaching

- 2.
Differentiate xy by using the product rule.

(3y

^{2}+ x) = -(2x + y)

**Example 2:**

Find for e^{x} + e^{y} = e^{x + y}
**Solution:**

e^{x} + e^{y} = e^{x + y}

Differentiate
both sides with respect to x

e^{x} + e^{y} = e^{(x + y)}
(x + y)
. Applying chain rule on the right side

e^{x} + e^{y} =
e^{(x + y)}(1 + )

Now factor out the term

( e^{y} - e^{(x + y)})
= e^{(x + y)} - e^{x}

Finally bring to one side of the equation.

**Summary:**

We differentiate the implicit functions by applying the following steps:

- Differentiate both sides of the equation with respect to x using the regular differentiation rules.
- Attach wherever we need to differentiate y term.
- Factor out the terms.
- Place the term on one side of the equation, bringing the remaining terms to the other side.

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