Questions: x^3 y - xy^3 = 4

Solution Steps

To find \(\frac{dy}{dx}\) using implicit differentiation, we will differentiate both sides of the equation with respect to \(x\). We will apply the product rule to the terms involving both \(x\) and \(y\), and remember that \(y\) is a function of \(x\), so when differentiating terms with \(y\), we will multiply by \(\frac{dy}{dx}\).

We start with the equation:

\[ x^{3} y - x y^{3} = 4 \]

To find \(\frac{dy}{dx}\), we differentiate both sides with respect to \(x\). Using the product rule, we differentiate the left-hand side:

\[ \frac{d}{dx}(x^{3} y) = 3x^{2} y + x^{3} \frac{dy}{dx} \] \[ \frac{d}{dx}(x y^{3}) = y^{3} + 3x y^{2} \frac{dy}{dx} \]

Thus, the differentiation gives us:

\[ 3x^{2} y + x^{3} \frac{dy}{dx} - (y^{3} + 3x y^{2} \frac{dy}{dx}) = 0 \]

Rearranging the differentiated equation, we have:

\[ 3x^{2} y - y^{3} + x^{3} \frac{dy}{dx} - 3x y^{2} \frac{dy}{dx} = 0 \]

This can be simplified to:

\[ ( x^{3} - 3x y^{2} ) \frac{dy}{dx} = - (3x^{2} y - y^{3}) \]

Now, we can isolate \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} = \frac{-(3x^{2} y - y^{3})}{x^{3} - 3x y^{2}} \]

This expression gives us the derivative \(\frac{dy}{dx}\) in terms of \(x\) and \(y\).

Final Answer