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The chain rule is a fundamental concept in calculus used to differentiate composite functions. In other words, it's a technique for finding the derivative of a function that is wrapped inside another function. This is important when dealing with implicit differentiation, as you often need to differentiate terms involving a variable that itself is a function of another variable.

For example, in the equation \(y^3 = \frac{{x-1}}{{x+1}}\), we recognize that \(y\) is a function of \(x\). Therefore, when differentiating \(y^3\) with respect to \(x\), we apply the chain rule: \[ \frac{{d}}{{dx}}(y^3) = 3y^2 \frac{{dy}}{{dx}} \]

Here’s a breakdown to understand it better:

• Differentiate the outer function: The outer function is \(y^3\), so the derivative is \( 3y^2 \).
• Multiply by the derivative of the inner function: Since \(y\) is a function of \(x\), its derivative is \( \frac{{dy}}{{dx}} \).

So, the complete differentiation using the chain rule is \[ 3y^2 \frac{{dy}}{{dx}} \], indicating the essential role of the chain rule in implicit differentiation.

The quotient rule is another crucial differentiation technique used when dealing with the ratio of two functions. Whenever you have a function divided by another, you will use the quotient rule to find its derivative.

For the function given by \( \frac{{x-1}}{{x+1}} \), we apply the quotient rule to differentiate it. The quotient rule states: \[\frac{{d}}{{dx}}\left( \frac{{u}}{{v}} \right) = \frac{{u'v - uv'}}{{v^2}} \]

Here, \(u = x-1\) and \(v = x+1\). Let’s apply the rule step by step:

• Calculate the derivative of the numerator: u' = 1
• Calculate the derivative of the denominator: v' = 1
• Apply the quotient rule formula: \[\frac{{(1)(x+1) - (x-1)(1)}}{{(x+1)^2}} = \frac{{x + 1 - (x-1)}}{{(x+1)^2}} = \frac{{2}}{{(x+1)^2}} \]

Thus, we find that the derivative of \(\frac{{x-1}}{{x+1}}\) using the quotient rule is \ \frac{{2}}{{(x+1)^2}} \. The quotient rule helps simplify the process of differentiating more complex fractions.

Finally, let's delve into derivatives. A derivative represents the rate of change of a function concerning one of its variables. In simpler terms, it's a measure of how a function's output value changes as the input value changes.

To differentiate implicitly and find \(\frac{{dy}}{{dx}}\), follow these steps:

• Differentiate both sides of the given equation with respect to \(x\).
• Use the chain rule for any terms involving \(y\) since \(y\) is a function of \(x\).
• Apply the quotient rule for differentiating a fraction like \(\frac{x-1}{x+1}\).
• Set the differentiated expressions equal.
• Solve for \(\frac{dy}{dx}\).

In the given problem, after differentiating both sides and applying the necessary rules, we found the derivative: \[ 3y^2 \frac{dy}{dx} = \frac{2}{(x+1)^2} \]
By isolating \(\frac{dy}{dx}\), we get: \[ \frac{dy}{dx} = \frac{2}{3y^2 (x+1)^2} \]
Derivatives are integral for understanding how functions change, and using rules like the chain and quotient rules simplifies the differentiation process.