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The chain rule is a fundamental concept used when differentiating composite functions. It becomes a vital tool in implicit differentiation, especially when dealing with expressions where one variable is dependent on another, like in our exercise. When you encounter a function like \( 4y^{3/2} \), and you know that \( y \) is a function of \( x \), you cannot differentiate it normally. Here, you apply the chain rule.

So, imagine you have a function \( u = y^{3/2} \). To find the derivative of this in terms of \( x \), first, differentiate \( u \) with respect to \( y \), then multiply the result by \( \frac{dy}{dx} \), which is the derivative of \( y \) with respect to \( x \). Essentially, the chain rule tells you:

• If \( u = y^{3/2} \), then \( \frac{d}{dy}[u] = \frac{3}{2} y^{1/2} \).
• Then, the derivative \( \frac{d}{dx}[4u] = 4 \cdot \frac{3}{2} y^{1/2} \cdot \frac{dy}{dx} \).

This step-by-step application is what gives the chain rule its power in solving implicit differentiation problems.

Derivative calculation involves finding out how a function changes as its input changes. In this context, with functions of \( x \) and \( y \), it becomes a bit more involved. Calculating the derivative of \( 4y^{3/2} \) requires understanding that \( y \) itself changes with \( x \).

Start by differentiating the constant multiplier with the chain rule:

• Differentiate \( 4y^{3/2} \) with respect to \( y \) to isolate the inner derivative \( 4 \cdot \frac{3}{2} y^{1/2} \).
• The factor of \( \frac{dy}{dx} \) accounts for the fact that \( y \) is actually changing.

This combination of moving parts is how you derive \( 6y^{1/2} \frac{dy}{dx} \). This simplification helps you understand how each part of the expression contributes to its rate of change.

In many math problems, especially those involving calculus, functions can be intertwined, with one depending on the other. A function of \( y \) like \( y^{3/2} \), where \( y \) itself is a function of \( x \), is a prime example of such cases. These are often tackled through implicit differentiation.

Implicit differentiation is employed here because \( y \) is not simple; it varies as \( x \) changes. Through this method, you acknowledge and handle the dependency without needing to express \( y \) solely in terms of \( x \). This approach helps calculate derivatives without rearranging equations extensively.

Consider the expression again: \( 4y^{3/2} \). We first acknowledge that \( y \) changes with \( x \) and differentiate accordingly. This approach elegantly solves problems where direct differentiation isn't straightforward due to interdependent variables, maintaining the integrity of the relationships between \( x \) and \( y \).