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When looking for the second derivative, you're finding how the rate of change of a function's slope (or first derivative) is itself changing. In other words, it measures the curvature of the graph. For example, if you have a graph that bends upwards, the second derivative will be positive. If the graph bends downwards, the second derivative will be negative.
The process begins once you've found the first derivative, in this case using implicit differentiation to get \( \frac{dy}{dx} = -\frac{x^2}{y^2} \). To get the second derivative, \( \frac{d^2y}{dx^2} \), you'll differentiate this result with respect to \( x \) again. This further differentiation reveals deeper insights into how the original curve changes as \( x \) changes.

The quotient rule is a very handy tool in calculus for finding the derivative of a function that is the division of two more basic functions. If you have a fraction where one function is divided by another, such as \( u(x) / v(x) \), the quotient rule states:

• The derivative of the entire fraction is \( \frac{v(x) \, u'(x) - u(x) \, v'(x)}{(v(x))^2} \).

In our solution, we used the quotient rule to differentiate \( \frac{-x^2}{y^2} \). This involves calculating each part:

• The derivative of the numerator \(-x^2\) is \( -2x \).
• The derivative of the denominator \(y^2\) requires applying the chain rule, as we'll discuss in the next section.

This method is crucial whenever you have a situation where one variable is influencing another in a fractional relationship. By using the quotient rule, we were able to find \( \frac{d^2 y}{dx^2} \) effectively.

The chain rule is a powerful method for differentiating compositions of functions. Essentially, it lets you take the derivative of a function that is nested within another function, or a function of a function. It reflects how a change in one variable propagates through a sequence of dependent changes.
In parts of the exercise, we encounter expressions like \( y^3 \) or \( y^2 \). Here, you can't directly differentiate because \( y \) is itself a dependent variable on \( x \). Thus, apply the chain rule. For instance:

• In differentiating \( y^3 \), you get \( 3y^2 \frac{dy}{dx} \), multiplying the power rule result by the derivative of \( y \) because \( y \) changes as \( x \) changes.

In the final derivative calculations, you'd see terms like \( 2y \frac{dy}{dx} \), which applies the chain rule by respecting the inner changes \( \frac{dy}{dx} \). Understanding this rule deeply helps unravel complex functions and interactions within different parts of mathematics.