Questions: Find (fracd yd x) by implicit differentiation. [ fracx^2x+y=y^2+7 fracd yd x=square ]

Solution Steps

We start with the equation

\[ \frac{x^{2}}{x+y} = y^{2} + 7. \]

To find \(\frac{dy}{dx}\), we differentiate the left-hand side with respect to \(x\):

\[ \frac{d}{dx}\left(\frac{x^{2}}{x+y}\right). \]

Using the quotient rule, we have:

\[ \frac{(x+y)(2x) - x^{2}(1 + \frac{dy}{dx})}{(x+y)^{2}}. \]

Next, we differentiate the right-hand side:

\[ \frac{d}{dx}(y^{2} + 7) = 2y \frac{dy}{dx}. \]

Now we set the derivatives from the left-hand side and the right-hand side equal to each other:

\[ \frac{(x+y)(2x) - x^{2}(1 + \frac{dy}{dx})}{(x+y)^{2}} = 2y \frac{dy}{dx}. \]

To isolate \(\frac{dy}{dx}\), we rearrange the equation:

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

This can be simplified to:

\[ \frac{dy}{dx} \left(2y (x+y)^{2} + x^{2}\right) = (x+y)(2x) - x^{2}. \]

Finally, we solve for \(\frac{dy}{dx}\):

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

Final Answer