91Ó°ÊÓ

The chain rule is a fundamental technique in calculus for differentiating composite functions. When you have a function within another function, you need to use the chain rule to find the derivative. The basic form of the chain rule can be written as \[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \].
This means you first perform the derivative of the outer function (keeping the inner function unchanged), and then multiply by the derivative of the inner function.
In our original exercise, we set the inner function to be \(u = \frac{4 - x^3}{x - x^2}\), and the outer function as \(f(u) = u^{\frac{1}{3}}\). Using the chain rule, we differentiate the outer function while keeping \(u\) the same, resulting in \( \frac{1}{3} u^{- \frac{2}{3}} \), and then multiply by the derivative of \(u\), which we will find using the quotient rule.

The quotient rule is used to differentiate functions that are divided by each other. If you have a function \( u(x) = \frac{a(x)}{b(x)} \), the quotient rule is given by \[ u' = \frac{a'(x) \, b(x) - a(x) \, b'(x)}{[b(x)]^2} \].
In our exercise, \(u = \frac{4 - x^3}{x - x^2}\). We need to differentiate the numerator \(a(x) = 4 - x^3\) and the denominator \(b(x) = x - x^2\). The derivatives are \(a'(x) = -3x^2\) and \(b'(x) = 1 - 2x\).
Plugging these into the quotient rule formula, we get:
\[ u' = \frac{(-3x^2)(x - x^2) - (4 - x^3)(1 - 2x)}{(x - x^2)^2} \]
Simplify this to:
\[ u' = \frac{4 - 4x + 2x^3 - x^4}{(x - x^2)^2} \]

Simplification is an essential step in calculus, especially after applying rules like the chain rule or quotient rule. Simplification makes your result cleaner and easier to interpret.
In our exercise, we simplified the original function \( f(x) = \sqrt[3]{\frac{4 - x^3}{x - x^2}} \) to \( f(x) = \left( \frac{4 - x^3}{x - x^2} \right)^{\frac{1}{3}} \). This allowed us to clearly see it as a composite function, making it easier to apply the chain and quotient rules.
After applying the chain rule and quotient rule, we further simplified the derivative of the inner function. The simplification steps involved combining like terms and reducing fractions where possible. This step makes the resulting derivative expression more manageable and clearer for interpretation.
The final combined derivative expression, after simplification, is:
\[ f'(x) = \frac{1}{3} \left( \frac{4 - x^3}{x - x^2} \right)^{- \frac{2}{3}} \cdot \left( \frac{4 - 4x + 2x^3 - x^4}{(x - x^2)^2} \right) \]