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In calculus, the chain rule is a formula for finding the derivative of a composition of functions. It is essential when dealing with composite functions, where one function is nested inside another. For example, if you have a function \(g(z) = [f(u(z))]^3\), you are dealing with a composite function made from an inner function \(u(z)\) and an outer function \(f(u) = u^3\).

To apply the chain rule, follow these steps:

• First, differentiate the outer function with respect to its argument. In this case, differentiate \(f(u) = u^3\) to get \(f'(u) = 3u^2\).
• Next, differentiate the inner function with respect to \(z\), which is \(u(z)\). This has been done in a previous step using the quotient rule.
• Finally, multiply the derivative of the outer function by the derivative of the inner function. This gives you the final derivative of the composite function.

The chain rule helps you break down complex differentiations into manageable steps, finding the rate at which the output of the composite function changes with respect to the input.

The power rule is one of the most fundamental techniques in calculus for finding derivatives. It is particularly useful when you need to find the derivative of polynomials or functions raised to a power.

When you have a function of the form \(u^n\), where \(n\) is a constant, the power rule states that the derivative is \(nu^{n-1}\). For example, in our function \(f(u) = u^3\), using the power rule gives us \(f'(u) = 3u^2\).

The power rule simplifies the process by providing a straightforward way to calculate the derivative of simple power functions, allowing you to quickly proceed to more complex rules like the chain rule when necessary. By recognizing when the power rule can be applied, you can effectively tackle polynomials of various degrees.

The quotient rule is a method used to differentiate functions that are the ratio of two differentiable functions. It is crucial when the function you need to differentiate is in the form of one function divided by another.

Given a function \(u(z) = \frac{z}{1+z^2}\), we apply the quotient rule to find its derivative. The quotient rule states that for a function \(\frac{f(z)}{g(z)}\), the derivative is:\[\frac{d}{dz}\left(\frac{f(z)}{g(z)}\right) = \frac{g(z)f'(z) - f(z)g'(z)}{[g(z)]^2}\]In our function, \(f(z) = z\) and \(g(z) = 1+z^2\). By applying the quotient rule, you find the derivative:\[\frac{1+z^2\cdot1 - z\cdot2z}{(1+z^2)^2} = \frac{1-z^2}{(1+z^2)^2}\]This step is essential in the calculation of the chain rule, where you first need the inner derivative before applying the outer function's derivative. The quotient rule provides a clear process to handle derivatives in fractional forms efficiently.