Find dy/dx: x³ + x²y + xy² + y³ = 81

Solution:

A derivative helps us to know the changing relationship between two variables. Consider the independent variable 'x' and the dependent variable 'y'.

The change in the value of the dependent variable with respect to the change in the value of the independent variable expression can be found using the derivative formula.

Given that:

x³ + x²y + x y ² + y³ = 81

On differentiating both sides wrt x , we get

⇒ d/dx (x³) + d/dx (x²y) + d/dx (x y ² ) = d/dx (81)

⇒ 3x² + [ y d/dx (x²) + x² (dy/dx) ]+ [ y² d/dx (x) + x d/dx (y²)] + 3y² dy/dx = 0

Since derivative of constant function is zero.

⇒ 3x² + [ y 2x + x² (dy/dx) ] + [ y² 1 + x 2y dy/dx] + 3y² dy/dx = 0

⇒ (x² + 2xy + 3y²) dy/dx + (3x² + 2xy + y²) = 0

Therefore,

dy/dx = -(3x² + 2xy + y²) / (x² + 2xy + 3y²)

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.3 Question 6

Find dy/dx: x³ + x²y + x y ² + y³ = 81

Summary:

The derivative of  x³ + x²y + x y ² + y³ = 81 with respect to x is dy/dx = - (3x² + 2xy + y²) / (x² + 2xy + 3y²) .A derivative helps us to know the changing relationship between two variables