A function is in the form y = f(x) is called explicit function.
For example y = x^{2} + 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 x^{2} + 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⁄dₓ. In such case differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.
In each case, find ᵈʸ⁄dₓ.
Example 1 :
x^{2} + y^{2} = 25
Solution :
x^{2} + y^{2} = 25
Differentiate both sides with respect to x.
Example 2 :
4x^{3}y^{2} + 3x = 1
Solution :
4x^{3}y^{2} + 3x = 1
Differentiate both sides with respect to x.
Example 3 :
x^{2}y^{2} + 3xy + y = 0
Solution :
x^{2}y^{2} + 3xy + y = 0
Differentiate both sides with respect to x.
Example 4 :
sin(x^{2}+ y^{2}) = e^{y}
Solution :
sin(x^{2}+ y^{2}) = e^{y}
Differentiate both sides with respect to x.
Example 5 :
ln(x^{2}+ y^{2}) = 3x
Solution :
ln(x^{2}+ y^{2}) = 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 :
x^{2} - 3ln y + y^{2} = 10
Solution :
x^{2} - 3ln y + y^{2} = 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 :
4x^{3} + ln y^{2} + 2y = 2x
Solution :
4x^{3} + ln y^{2} + 2y = 2x
Using the properties of logarithms,
4x^{3} + 2ln y + 2y = 2x
Differentiate both sides with respect to x.
Example 10 :
4xy + ln x^{2}y = 7
Solution :
4xy + ln x^{2}y = 7
Using the properties of logarithms,
4xy + ln x^{2 }+ ln y = 7
4xy + 2ln x^{ }+ ln y = 7
Differentiate both sides with respect to x.
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