We have worked with functions where y can be expressed explicitly in terms of x.
For example, let’s take a look at the following functions

1. x + y = 2 which can be written as y = 2 – x.

2. y = x²

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. x² + y³ + xy = 2

2. tan(x + y) – tan(x – y) = 1

As the above functions cannot be explicitly written in terms of x we call them implicit functions.

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

Example 1:

Find when x² + y³ + xy = 16

Solution:

Given: x² + y³ + xy = 16

Differentiating both sides with respect to x we get,

2x + 3y² + x + y(1) = 0

Note:

1. We differentiated y³ by using the power rule and attaching

2. Differentiate xy by using the product rule.

Next, bring to one side of the equation as follows:

(3y² + 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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