Questions: Directions: Use the equation given below to answer the following question. x^4 y^2 - x^4 y + 2 x y^3 = 0 Derivative Use implicit differentiation to calculate d x / d y. Note that we are calculating the derivative of x with respect to y (the opposite of what we normally do). - This means we need to regard y as the independent variable and z as the dependent variable. d x / d y =

Solution Steps

We start with the equation given in the problem:

\[ x^{4} y^{2} - x^{4} y + 2 x y^{3} = 0 \]

We differentiate both sides with respect to \(y\):

\[ \frac{d}{dy}(x^{4} y^{2}) - \frac{d}{dy}(x^{4} y) + \frac{d}{dy}(2 x y^{3}) = 0 \]

Applying the product rule and chain rule, we find:

\[ 2 x^{4} y - x^{4} + 6 x y^{2} = 0 \]

Next, we rearrange the differentiated equation to isolate \(\frac{d x}{d y}\):

\[ 2 x^{4} y + 6 x y^{2} = x^{4} \]

This can be rewritten as:

\[ \frac{d x}{d y} = \frac{x^{4} - 2 x^{4} y - 6 x y^{2}}{0} \]

However, since we are looking for \(\frac{d x}{d y}\) explicitly, we note that the expression simplifies to:

\[ \frac{d x}{d y} = \frac{x^{4} - 2 x^{4} y - 6 x y^{2}}{0} \]

Final Answer