91Ó°ÊÓ

When solving calculus problems involving implicit differentiation, it's essential to know how to take the derivative with respect to a specific variable like \( x \). This means you treat every other variable (such as \( y \)) as implicitly dependent on \( x \). For the equation \( 3x + 2y = 5 \), we differentiate each term separately.

• The term \( 3x \) is straightforward because it's directly in terms of \( x \).
• The term \( 2y \) requires more attention since \( y \) is not originally a function of \( x \).

By taking the derivative with respect to \( x \), you begin the process of uncovering how changes in \( x \) affect the entire equation.

The chain rule is a key tool used in implicit differentiation. It helps us deal with derivatives when one function is nested inside another. With implicit differentiation, it's useful for terms where \( y \) depends on \( x \). When we differentiate such terms, we must account for this dependency by using the chain rule.When we differentiate \( 2y \), we apply the chain rule: - The derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} \). - Thus, the derivative of \( 2y \) becomes \( 2 \cdot \frac{dy}{dx} \).
Using the chain rule ensures that we're capturing the hidden relationships between \( y \) and \( x \), which is crucial for accurate differentiation in implicit contexts.

The power rule simplifies the process of differentiation when dealing with polynomial terms. For any term \( ax^n \), the derivative is \( anx^{n-1} \). In our exercise, the term \( 3x \) is effectively \( 3x^1 \), making its differentiation very straightforward.Applying the power rule: - For \( 3x \), the derivative with respect to \( x \) is simply \( 3 \), as \( n = 1 \) leads to \( 1 \times 3x^{1-1} = 3 \times x^0 = 3 \).This rule is the reason our calculations were so simple for this part of the problem, and it often makes differentiating polynomial terms intuitive and quick.

Once differentiation is complete, the next step involves solving for \( \frac{dy}{dx} \). We need to rearrange the equation to isolate this term. In the equation \( 3 + 2 \frac{dy}{dx} = 0 \), our goal is to solve for \( \frac{dy}{dx} \). The process involves:

• Subtracting 3 from both sides, resulting in \( 2 \frac{dy}{dx} = -3 \).
• Dividing both sides by 2 to isolate \( \frac{dy}{dx} \).

Thus, the derivative \( \frac{dy}{dx} \) is \( -\frac{3}{2} \). This solution tells us the rate of change of \( y \) with respect to \( x \) for the given equation. Solving for \( \frac{dy}{dx} \) is usually the ultimate goal in problems involving implicit differentiation, as it gives insight into the relationship between the variables.