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When we talk about finding the first derivative in a parametric context, we're focusing on determining how one variable changes with respect to another as they are both defined in terms of a parameter. In the case of our example, we are given two functions: \( x = \cos \phi \) and \( y = 3\sin \phi \). Here, \( \phi \) is the parameter. To find the first derivative \( \frac{dy}{dx} \), we follow a specific process:

• First, differentiate \( x \) with respect to \( \phi \): \( \frac{dx}{d\phi} = -\sin \phi \).
• Next, differentiate \( y \) with respect to \( \phi \): \( \frac{dy}{d\phi} = 3 \cos \phi \).
• Finally, apply the chain rule to combine these results and find \( \frac{dy}{dx} = \frac{dy/d\phi}{dx/d\phi} \).

This gives you \( \frac{dy}{dx} = -3 \cot \phi \), which tells us the slope of the curve or how \( y \) changes in relation to \( x \) for any given \( \phi \). By substituting \( \phi = \frac{5\pi}{6} \), calculation shows \( \frac{dy}{dx} \) is \( 3\sqrt{3} \), illustrating the steepness of the curve at that particular parameter value.

Calculating the second derivative, \( \frac{d^2 y}{dx^2} \), provides insights into the concavity of the curve at a specific point, which tells us how the rate of change itself is changing. This can be important for understanding the curvature of the graph.To find \( \frac{d^2 y}{dx^2} \), use the first derivative expression found earlier. You differentiate this first derivative with respect to \( \phi \), and then multiply by \( \frac{d\phi}{dx} \), as shown in the solution steps:

• Start with the derivative of \(-3 \cot \phi\). The differentiation yields \( 3 \csc^2 \phi \).
• Then, multiply by \( \frac{d\phi}{dx} = -\frac{1}{\sin \phi} \) to account for the change in \( \phi \) with \( x \).

Plugging in \( \phi = \frac{5\pi}{6} \), this gives us \( \frac{d^2 y}{dx^2} = -48 \), meaning the curve is concave downward at that specific point. Calculating the second derivative in this manner helps determine the nature of the curvature, giving a complete view of the graph's behavior.

The chain rule is a fundamental technique in calculus that allows us to differentiate composite functions. In parametric differentiation, it is especially useful because we often deal with functions of a parameter, not directly expressed in terms of each other. Therefore, the chain rule gives us a tool to relate these functions.In our example:

• We have \( y = 3\sin \phi \) and \( x = \cos \phi \), both in terms of \( \phi \).
• To find \( \frac{dy}{dx} \), which is the derivative of \( y \) with respect to \( x \), we apply the chain rule as: \( \frac{dy}{dx} = \frac{dy/d\phi}{dx/d\phi} \).

Using the derivatives \( \frac{dy}{d\phi} = 3 \cos \phi \) and \( \frac{dx}{d\phi} = -\sin \phi \), the chain rule helps us express the rate at which \( y \) changes with respect to \( x \) as a single expression. This is crucial for solving problems where direct differentiation isn't possible without breaking down the problem into simpler parts.