IMPLICIT DIFFERENTIATION

A function is in the form yf(x) is called explicit function.

For example y = x2 + 2x + 3, where y is directly defined as a function of x.

A function is in the form f(x, y) = 0 is called implicit function.

For example x2 + 2xy = 0, where y is not directly defined as a function of x.

In case of implicit functions, if y be a differentiable function of x, no attempt is required to express y as an explicit function of x for finding out ᵈʸ⁄d. In such case differentiate both sides with respect to x and solve for ᵈʸ⁄d.

In each case, find ᵈʸ⁄d.

Example 1 :

x2 + y2 = 25

Solution :

x2 + y2 = 25

Differentiate both sides with respect to x.

Example 2 :

4x3y2 + 3x = 1

Solution :

4x3y2 + 3x = 1

Differentiate both sides with respect to x.

Example 3 :

x2y2 + 3xy + y = 0

Solution :

x2y2 + 3xy + y = 0

Differentiate both sides with respect to x.

Example 4 :

sin(x2+ y2) = ey

Solution :

sin(x2+ y2) = ey

Differentiate both sides with respect to x.

Example 5 :

ln(x2+ y2) = 3x

Solution :

ln(x2+ y2) = 3x

Differentiate both sides with respect to x.

Example 6 :

esiny = 2xy + 5

Solution :

esiny = 2xy + 5

Differentiate both sides with respect to x.

Example 7 :

x2 - 3ln y + y2 = 10

Solution :

x2 - 3ln y + y2 = 10

Differentiate both sides with respect to x.

Example 8 :

ln xy + 5x = 30

Solution :

ln xy + 5x = 30

Using the properties of logarithms,

ln x + ln y + 5x = 30

Differentiate both sides with respect to x.

Example 9 :

4x3 + ln y2 + 2y = 2x

Solution :

4x3 + ln y2 + 2y = 2x

Using the properties of logarithms,

4x3 + 2ln y + 2y = 2x

Differentiate both sides with respect to x.

Example 10 :

4xy + ln x2y = 7

Solution :

4xy + ln x2y = 7

Using the properties of logarithms,

4xy + ln x+ ln y = 7

4xy + 2ln x + ln y = 7

Differentiate both sides with respect to x.

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