Suppose that $$ x $$ and $$ y $$ are both differentiable functions of $$ t $$ and are related by the given equation. Use implicit differentiation with respect to $$ t $$ to determine $$ \frac{d y}{d t} $$ in terms of $$ x, y $$, and $$ \frac{d x}{d t} $$. $$ 8 x y-8 x^{3}=11 $$ $$ \frac{d y}{d t}=\square \frac{d x}{d t} $$

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To find $$\frac{d y}{d t}$$ using implicit differentiation, we start with the given equation:

$$
8xy - 8x^3 = 11
$$

We will differentiate both sides of this equation with respect to $$t$$.

1. Differentiate the left side:
   - For $$8xy$$, we use the product rule:
     $$
     \frac{d}{dt}(8xy) = 8\left(x \frac{dy}{dt} + y \frac{dx}{dt}\right)
     $$
   - For $$-8x^3$$, we use the chain rule:
     $$
     \frac{d}{dt}(-8x^3) = -24x^2 \frac{dx}{dt}
     $$

2. Differentiate the right side:
   - The derivative of a constant (11) is 0:
     $$
     \frac{d}{dt}(11) = 0
     $$

Putting it all together, we have:

$$
8\left(x \frac{dy}{dt} + y \frac{dx}{dt}\right) - 24x^2 \frac{dx}{dt} = 0
$$

Now, we can simplify this equation:

$$
8x \frac{dy}{dt} + 8y \frac{dx}{dt} - 24x^2 \frac{dx}{dt} = 0
$$

Next, we can isolate $$\frac{dy}{dt}$$:

$$
8x \frac{dy}{dt} = -8y \frac{dx}{dt} + 24x^2 \frac{dx}{dt}
$$

Factoring out $$\frac{dx}{dt}$$ from the right side:

$$
8x \frac{dy}{dt} = \left(24x^2 - 8y\right) \frac{dx}{dt}
$$

Now, divide both sides by $$8x$$ (assuming $$x \neq 0$$):

$$
\frac{dy}{dt} = \frac{24x^2 - 8y}{8x} \frac{dx}{dt}
$$

Simplifying the fraction on the right:

$$
\frac{dy}{dt} = \left(3x - \frac{y}{x}\right) \frac{dx}{dt}
$$

Thus, we can express $$\frac{dy}{dt}$$ in terms of $$x$$, $$y$$, and $$\frac{dx}{dt}$$:

$$
\frac{dy}{dt} = \left(3x - \frac{y}{x}\right) \frac{dx}{dt}
$$

So, the final answer is:

$$
\frac{dy}{dt} = \left(3x - \frac{y}{x}\right) \frac{dx}{dt}
$$
$$ \frac{dy}{dt} = \left(3x - \frac{y}{x}\right) \frac{dx}{dt} $$

Suppose that and are both differentiable functions of and are related by the given
equation. Use implicit differentiation with respect to to determine in terms of , and
.

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Suppose that and are both differentiable functions of and are related by the given equation. Use implicit differentiation with respect to to determine in terms of , and .