Implicit Differentiation

IMPLICIT DIFFERENTIATION

So far, we have seen differentiation of function given by equations of the form y = f(x), where y is written explicitly in terms of x. The function is then said to be in explicit form. But in practical situations we come across equations such as x3 + y3 = 3xy, where y cannot be written in terms of x explicitly. Such an equation is in the form of F(x, y) = 0, where y is defined implicitly as a function of x, is said to be in implicit form. To differentiate the functions in implicit form, we treat y as a function of x and differentiate with respect to x. This type of differentiation is known as implicit differentiation.

This technique is illustrated in the following examples.

WORKED EXAMPLES

Example 1 

Solution

Given

x2 + xy − y2 = 10        (1)

It is implicite function.

⸫ differentiating w. r. to x, we get

Example 2 

Solution

Example 3 

Find y' and y" if x4 + y4 = 4

Solution 

Given x4 + y4 = 4        (1)

It is implicit function

Differentiating (1) w. r. to x, we get

Differentiating (2) w. r. to x, we get

From (2), we get

Substituting in (3), we get

Example 4 

Solution

Given the implicit form

x2 - y2 + sin (xy) = 0

Differentiating with respect to x, treating y as a function of x, we get

Example 5 

Solution

Given the implicit form

x3 + y3 = 6xy

Differentiating with respect to x, treating y as a function of x, we get

Dividing by 3, we get

Example 6 

Solution

Given the implicit form

sin y = x sin(a + y)

Differentiating with respect to y, we get

Example 7 

If x2 + xy + y = 1, then find the value of ym at the point where x = 1

Solution 

From (3), we get

⇒ 2 + 1⸱ yn + 2(-2) + 0 + 0 = 0

⇒ yn – 2 = 0 ⇒ yn = 2

From (4), we get

Example 8 

Solution 

Example 9 

Solution 

Example 10 

Solution 

Given the implicit form

Differentiating with respect to x, we get