91Ó°ÊÓ

The second derivative, noted as \(\frac{d^{2} y}{d x^{2}}\), is the derivative of the derivative of a function. It provides information on the concavity and the rate of change of the slope of the graph of the function. Finding this derivative implicitly involves some additional steps because we're working with an equation where \(y\) is not isolated. This method allows us to differentiate equations that define \(y\) implicitly in terms of \(x\).
To find the second derivative for the function \(x^2 - y^2 = 36\), we first differentiate both terms with respect to \(x\) to find \(y'\), and then do it again for \(y''\). The main idea is to uncover how the slopes themselves change, capturing more nuanced features of the curve.

• The first differentiation yields \(y' = \frac{x}{y}\), which is the slope at any point.
• Calculating \(y''\) gives the curve's concavity or nature of curve turning, yielded in the form \(y'' = \frac{y^2 - 2x^2}{y^3}\), later simplified as \(y'' = -\frac{x^2 + 36}{y^3}\).

The quotient rule is crucial for differentiating expressions where one function divides another. It states that for a function \(f(x) = \frac{g(x)}{h(x)}\), the derivative \(f'(x)\) is \(\frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\). When handling \(y' = \frac{x}{y}\), we need to differentiate it again to find \(y''\).
Using the quotient rule involves several steps:

• Identify \(g(x) = x\) and \(h(x) = y\), then find their derivatives: \(g'(x) = 1\) and \(h'(x)\) is found using the chain rule.
• Plug into the quotient rule: \(y'' = \frac{(1)\cdot y - x \cdot y'}{y^2}\).
• Simplifying this expression incorporates \(y' = \frac{x}{y}\), further detailing the changes in the function.

The quotient rule thus supports finding higher derivatives and enables handling more complex functions.

The chain rule is a fundamental tool in calculus for differentiating composite functions. It aids in understanding how changes in one variable affect another variable through a linked process.
In the context of implicit differentiation, it’s especially useful. When differentiating \(y^2 = 36 - x^2\) implicitly, we see how both \(x\) and \(y\) interact through this rule. With the chain rule:

• When taking the derivative of \(y^2\), treat \(y\) as a function of \(x\), leading to \(2y \cdot y' = -2x\).
• This step uncovers \(y'\) as part of the change in \(y\) with respect to \(x\).

The chain rule also interacts with the quotient rule later in the differentiation process, underscoring its versatility and necessity in calculus.