Solution:

Step 1:

Apply the derivative operator to both sides of the given equation: $\frac{d}{dx}(xy) = \frac{d}{dx}(12)$.

Step 2:

Take the derivative of the left-hand side.

Step 2.1:

Utilize the Product Rule for differentiation: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$, where $u = x$ and $v = y$. This yields $x\frac{d}{dx}(y) + y\frac{d}{dx}(x)$.

Step 2.2:

Express $\frac{d}{dx}(y)$ as $dy/dx$. The equation becomes $x\frac{dy}{dx} + y\frac{d}{dx}(x)$.

Step 2.3:

Apply the Power Rule for differentiation: $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n = 1$. This simplifies to $x\frac{dy}{dx} + y \cdot 1$.

Step 2.4:

Multiply $y$ by $1$ to maintain the equation: $x\frac{dy}{dx} + y$.

Step 3:

Differentiate the right-hand side, which is a constant. The derivative of a constant is $0$: $0$.

Step 4:

Combine the results of the differentiation to form the new equation: $x\frac{dy}{dx} + y = 0$.

Step 5:

Isolate $\frac{dy}{dx}$.

Step 5.1:

Subtract $y$ from both sides to get $x\frac{dy}{dx} = -y$.

Step 5.2:

Divide the equation by $x$ to solve for $\frac{dy}{dx}$.

Step 5.2.1:

Divide both sides by $x$: $\frac{x\frac{dy}{dx}}{x} = \frac{-y}{x}$.

Step 5.2.2:

Simplify the left-hand side.

Step 5.2.2.1:

Eliminate the common $x$ factor: $\frac{\cancel{x}\frac{dy}{dx}}{\cancel{x}} = \frac{-y}{x}$.

Step 5.2.2.2:

Simplify to get $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{-y}{x}$.

Step 5.2.3:

Simplify the right-hand side, if necessary.

Step 5.2.3.1:

Position the negative sign in front of the fraction: $\frac{dy}{dx} = -\frac{y}{x}$.

Step 6:

Replace $\frac{dy}{dx}$ with the derivative notation: $\frac{dy}{dx} = -\frac{y}{x}$.

Knowledge Notes:

The problem-solving process involves applying differential calculus to find the derivative of a function with respect to a variable. The key knowledge points involved in this process include:

1. Derivative Operator: The notation $\frac{d}{dx}$ represents the derivative of a function with respect to the variable $x$.

2. Product Rule: A rule used when differentiating products of two functions, which states that $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$.

3. Power Rule: A basic rule for differentiation, which states that for any real number $n$, the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.

4. Constant Rule: The derivative of a constant is zero. This is because constants do not change, so their rate of change with respect to any variable is zero.

5. Solving for a Derivative: After applying the rules of differentiation, algebraic manipulation is often required to isolate the derivative term ($\frac{dy}{dx}$ in this case).

6. Notation: The notation $\frac{dy}{dx}$ represents the derivative of $y$ with respect to $x$, which is what we are solving for in this problem.

Understanding these concepts is crucial for successfully applying differentiation to solve problems in calculus.