91Ó°ÊÓ

Implicit differentiation often uses the chain rule, which is a technique that helps us differentiate composite functions. When dealing with equations that involve both \(x\) and \(y\), like \(x^{2/3} + y^{2/3} = 1\), the chain rule becomes essential for handling the derivative of terms with \(y\).
The chain rule states that the derivative of a composite function \(f(g(x))\) is \(f'(g(x)) \cdot g'(x)\). In the context of our exercise, when differentiating \(y^{2/3}\), we treat \(y\) as a function of \(x\). This means:

• The derivative of \(y^{2/3}\) with respect to \(y\) is \(\frac{2}{3}y^{-1/3}\).
• Since \(y\) is a function of \(x\), we multiply by \(\frac{dy}{dx}\) (the derivative of \(y\) with respect to \(x\)).

This gives us \(\frac{2}{3}y^{-1/3} \cdot \frac{dy}{dx}\) for the derivative of \(y^{2/3}\).
The chain rule allows us to account for the hidden dependency on \(x\) through \(y\) and is a powerful tool in implicit differentiation.

The power rule simplifies differentiation of terms that are in the form of a variable raised to a constant power, making it incredibly handy in calculus. The power rule states that if you have a function \(x^n\), then its derivative is \(nx^{n-1}\). We apply this rule to both parts of the given equation: \(x^{2/3} + y^{2/3} = 1\).
Let's break down each term:

• For \(x^{2/3}\), using the power rule, the derivative is \(\frac{2}{3}x^{-1/3}\).
• For \(y^{2/3}\), while using the power rule gives us \(\frac{2}{3}y^{-1/3}\), we also need to apply the chain rule because \(y\) is a function of \(x\). This adds the \(\frac{dy}{dx}\) term to the derivative, yielding \(\frac{2}{3}y^{-1/3} \cdot \frac{dy}{dx}\).

By combining the power rule with other differentiation techniques, like the chain rule, we can handle more complex differential problems that come up when using implicit differentiation.

Once we have the first derivative \(\frac{dy}{dx}\), finding the second derivative \(\frac{d^2y}{dx^2}\) involves differentiating the first derivative again. This process helps reveal how the rate of change is itself changing. After obtaining \(\frac{dy}{dx} = -\left(\frac{x}{y}\right)^{1/3}\), we want to find \(\frac{d^2y}{dx^2}\) by differentiating this result.
The differentiation of \(-\left(\frac{x}{y}\right)^{1/3}\) employs the chain rule and the quotient rule. The quotient rule is applied here as we're dealing with the fraction \(\frac{x}{y}\):

• Calculate \(\frac{d}{dx}\left(\frac{x}{y}\right) = \frac{y - x \cdot \frac{dy}{dx}}{y^2}\).

Bringing in the chain rule, you differentiate \(-\left(\frac{x}{y}\right)^{1/3}\) with respect to the result above, forming:

• \(\frac{d^2y}{dx^2} = -\frac{1}{3}\left(\frac{x}{y}\right)^{-2/3} \cdot \frac{y - x \frac{dy}{dx}}{y^2}\).

Substituting the value of \(\frac{dy}{dx}\) simplifies it to the shown result in the original solution. This step-by-step progression through differentiation shows how the relationship between \(x\) and \(y\) evolves beyond its initial change, offering deeper insights into the function's behavior.