IMPLICIT DIFFERENTIATION

In each case, find ᵈʸ⁄dₓ.

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) = e^y

Solution :

sin(x²+ y²) = e^y

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 :

e^siny = 2xy + 5

Solution :

e^siny = 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 x² + ln y = 7

4xy + 2ln x + ln y = 7

Differentiate both sides with respect to x.

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