Find $$ \frac{d y}{d x} $$ by implicit differentiation. $$ 3 x^{2}+7 x y-y^{2}=8 $$ $$ \frac{d y}{d x}=\square $$

Question

UpStudy AI · Accepted Answer

We start with the equation

$$
3x^2 + 7xy - y^2 = 8.
$$

Differentiate both sides with respect to $$ x $$.

1. Differentiate $$ 3x^2 $$:
   $$
   \frac{d}{dx}(3x^2) = 6x.
   $$

2. Differentiate $$ 7xy $$ (using the product rule):
   $$
   \frac{d}{dx}(7xy) = 7\left(y + x \frac{dy}{dx}\right) = 7y + 7x\frac{dy}{dx}.
   $$

3. Differentiate $$ -y^2 $$ (using the chain rule):
   $$
   \frac{d}{dx}(-y^2) = -2y\frac{dy}{dx}.
   $$

Now, differentiate the right-hand side:
$$
\frac{d}{dx}(8) = 0.
$$

Putting it all together, we have:
$$
6x + 7y + 7x\frac{dy}{dx} - 2y\frac{dy}{dx} = 0.
$$

Group the terms with $$\frac{dy}{dx}$$ on one side:
$$
7x\frac{dy}{dx} - 2y\frac{dy}{dx} = -6x - 7y.
$$

Factor out $$\frac{dy}{dx}$$:
$$
\frac{dy}{dx}(7x - 2y) = -6x - 7y.
$$

Finally, solve for $$\frac{dy}{dx}$$:
$$
\frac{dy}{dx} = \frac{-6x - 7y}{7x - 2y}.
$$

Thus, the derivative is

$$
\boxed{\frac{dy}{dx} = \frac{-6x - 7y}{7x - 2y}}.
$$
$$ \frac{dy}{dx} = \frac{-6x - 7y}{7x - 2y} $$

Find \( \frac{d y}{d x} \) by implicit differentiation. \[ 3 x^{2}+7 x y-y^{2}=8 \] \( \frac{d y}{d x}=\square \)