Find d y / d x and d^2 y / d x^2, and find the slope and...

Find $d y / d x$ and $d^{2} y / d x^{2}$, and find the slope and concavity (if possible) at the given value of the parameter.
$$
\begin{array}{ll}
\underline{\text { Parametric Equations }} & \underline{\text { Point }} \\
x=\cos \theta, y=3 \sin \theta &\quad \theta=0
\end{array}
$$

Key Concepts

Chain Rule

This fundamental rule in calculus is used to differentiate composite functions. In the context of parametric equations, the chain rule allows us to relate the derivatives with respect to the parameter to the derivative of y with respect to x, making it essential for obtaining both the first and second derivatives.

Second Derivative (Concavity Analysis)

The second derivative d²y/dx² in the context of parametric equations provides information about the concavity of the curve. It is calculated by differentiating the first derivative with respect to the parameter and then dividing by dx/d?, i.e., d²y/dx² = (d/d? (dy/dx)) / (dx/d?). This derivative indicates whether the curve is concave up or concave down at a point.

First Derivative (Slope of the Tangent)

The first derivative at a point on a parametric curve gives the slope of the tangent line. It is determined by employing the formula dy/dx = (dy/d?) / (dx/d?), which is a direct application of the chain rule and expresses how y changes with respect to x as the parameter changes.

Parametric Differentiation

This is the process of finding derivatives when the variables are defined in terms of a third variable, often a parameter like t or ?. Instead of y being directly a function of x, both x and y are functions of the parameter, and the derivative dy/dx is obtained by dividing dy/d? by dx/d?.

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