Differentiation Techniques: Derivative of ( x^2 frac{x^2 y}{x - 2y} )

In this article, we will be exploring the differentiation of an implicit function and solving for its derivative. The function we are examining is ( x^2 frac{x^2 y}{x - 2y} ). We will use implicit differentiation to find the derivative with respect to ( x ). This approach is a fundamental technique in calculus and is essential for understanding complex functions.

Introduction to Implicit Differentiation

Implicit differentiation is a method used to find the derivative of a function that is defined implicitly by an equation involving both ( x ) and ( y ). It is particularly useful when the function cannot be easily solved for ( y ).

First Attempt at Differentiation

Let's start by solving the given equation for the derivative of ( y ) with respect to ( x ): [ x^2 frac{x^2 y}{x - 2y} ]

We will differentiate both sides with respect to ( x ). First, we can rewrite the equation:

[frac{d}{dx} [x^2] frac{d}{dx} left[ frac{x^2 y}{x - 2y} right]]

Using the power rule, we know that (frac{d}{dx} [x^2] 2x). Let's proceed with the right-hand side using implicit differentiation:

[frac{d}{dx} left[ frac{x^2 y}{x - 2y} right] frac{(x - 2y) cdot frac{d}{dx} [x^2 y] - x^2 y cdot frac{d}{dx} [x - 2y]}{(x - 2y)^2}]

Next, we differentiate the numerator and the denominator term by term:

[frac{d}{dx} [x^2 y] 2xy x^2 frac{dy}{dx}] [frac{d}{dx} [x - 2y] 1 - 2 frac{dy}{dx}]

Substituting these into our equation:

[frac{(x - 2y) cdot (2xy x^2 frac{dy}{dx}) - x^2 y cdot (1 - 2 frac{dy}{dx})}{(x - 2y)^2}]

Simplifying the numerator:

[(x - 2y)(2xy x^2 frac{dy}{dx}) - x^2 y (1 - 2 frac{dy}{dx}) 2x^2 y - 4xy^2 x^3 frac{dy}{dx} - 2x^2 y frac{dy}{dx} - x^2 y 2x^2 y frac{dy}{dx}]

Simplifying further:

[-x^2 y - 4xy^2 x^3 frac{dy}{dx}]

Thus, the expression becomes:

[frac{-x^2 y - 4xy^2 x^3 frac{dy}{dx}}{(x - 2y)^2} 2x]

Multiplying both sides by the denominator to isolate ( frac{dy}{dx} ):

[ -x^2 y - 4xy^2 x^3 frac{dy}{dx} 2x(x - 2y)^2 ]

Solving for ( frac{dy}{dx} ):

[ x^3 frac{dy}{dx} 2x(x - 2y)^2 x^2 y 4xy^2 ]

Dividing both sides by ( x^3 ):

[ frac{dy}{dx} frac{2x(x - 2y)^2 x^2 y 4xy^2}{x^3} ]

Second Attempt at Differentiation

We can also approach this by directly differentiating the equation: [ x^2 frac{x^2 y}{x - 2y} ]

Let's rewrite it and differentiate:

[frac{d}{dx} [x^2] frac{d}{dx} left[ frac{x^2 y}{x - 2y} right]]

Using implicit differentiation:

[ 2x frac{(x - 2y) cdot (2xy x^2 frac{dy}{dx}) - x^2 y (1 - 2 frac{dy}{dx})}{(x - 2y)^2} ]

Further simplifying:

[ 2x frac{-x^2 y - 4xy^2 x^3 frac{dy}{dx}}{(x - 2y)^2} ]

This leads to the same expression as before.

Final Simplified Solution

By solving step by step, we find:

[frac{dy}{dx} frac{3x^2 - 4xy - 1}{21x^2} ]

Thus, the derivative of ( x^2 ) with respect to ( y ) from the given function is (boxed{frac{3x^2 - 4xy - 1}{21x^2}}).

Conclusion

Through the process of implicit differentiation, we successfully found the derivative of ( x^2 frac{x^2 y}{x - 2y} ). This approach is valuable for solving more complex and implicit functions in calculus.