Question:

Find the derivative $\frac{\mathrm{d}y}{\mathrm{d}x}$ of the equation $x^2 +xy + y^2 = 9$ using implicit differentiation.

Answer:

To find the derivative $\frac{\mathrm{d}y}{\mathrm{d}x}$ using implicit differentiation, we will differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$.

First, we differentiate each term on the left side of the equation using the chain rule:

• Taking the derivative of $x^2$ with respect to $x$ gives $2x$.
• Taking the derivative of $xy$ with respect to $x$ yields $y + x\frac{\mathrm{d}y}{\mathrm{d}x}$.
• Taking the derivative of $y^2$ with respect to $x$ gives $2y\frac{\mathrm{d}y}{\mathrm{d}x}$.

Next, we apply the chain rule when differentiating the constant term on the right side of the equation. Since $9$ is a constant, its derivative is $0$.

Putting it all together, the equation becomes:

We can now isolate $\frac{\mathrm{d}y}{\mathrm{d}x}$ by collecting the terms that include the derivative on one side:

Factoring out $\frac{\mathrm{d}y}{\mathrm{d}x}$:

Finally, we get the derivative $\frac{\mathrm{d}y}{\mathrm{d}x}$ by dividing both sides of the equation by $(x + 2y)$:

Thus, the derivative of the equation $x^2 + xy + y^2 = 9$ with respect to $x$ is $\frac{-2x-y}{x+2y}$.