Problem 40 - Calculus and its applications

Question

Differentiate each function $$ f(x)=\sqrt[3]{\frac{4-x^{3}}{x-x^{2}}} $$

Step-by-Step Solution

Answer

Differentiate using the chain and quotient rules.

Step 1: Understanding the Function

The function given is \( f(x) = \sqrt[3]{\frac{4-x^3}{x-x^2}} \). This means we need to differentiate the cube root of a quotient.

Step 2: Apply the Chain Rule

To differentiate \( f(x) = \sqrt[3]{u} \) where \( u = \frac{4-x^3}{x-x^2} \), use the chain rule: \( f'(x) = \frac{1}{3} u^{-rac{2}{3}} \cdot u'(x) \). First find the derivative of \( u \).

Step 3: Differentiate the Inner Function u(x)

The function \( u(x) = \frac{4-x^3}{x-x^2} \) is a quotient, so apply the quotient rule: \( u'(x) = \frac{(v \cdot u' - u \cdot v')}{v^2} \) where \( u = 4-x^3 \) and \( v = x-x^2 \).

Step 4: Compute Derivatives of u and v

For \( u = 4-x^3 \), \( u' = -3x^2 \). For \( v = x-x^2 \), \( v' = 1-2x \).

Step 5: Apply the Quotient Rule

Substitute into the quotient rule formula: \( u'(x) = \frac{((x-x^2)(-3x^2) - (4-x^3)(1-2x))}{(x-x^2)^2} \). Simplify this expression.

Step 6: Simplify the Derivative

Expanding the terms: - \( (x-x^2)(-3x^2) = -3x^3 + 3x^4 \) - \( (4-x^3)(1-2x) = 4 - 8x - x^3 + 2x^4 \) Combine these: \( u'(x) = \frac{-3x^3 + 3x^4 - 4 + 8x + x^3 - 2x^4}{(x-x^2)^2} \).

Step 7: Simplify Further

Combine like terms: \( u'(x) = \frac{-x^3 + x^4 + 8x - 4}{(x-x^2)^2} \).

Step 8: Substitute Back to Find f'(x)

Now substitute back into the chain rule expression: \( f'(x) = \frac{1}{3} \left(\frac{4-x^3}{x-x^2}\right)^{-rac{2}{3}} \cdot \frac{-x^3 + x^4 + 8x - 4}{(x-x^2)^2} \).

Step 9: Final Simplification

Factor \( (x-x^2)^{-4/3} \) from \( u \) expression, if needed, and simplify the result for \( f'(x) \) if possible.

Key Concepts

Quotient Rule

The Quotient Rule is a special technique in calculus used to differentiate functions that are divided by each other, more formally known as a quotient of two functions. When faced with a function like \( u(x) = \frac{4-x^3}{x-x^2} \), which is structured as one expression divided by another, the Quotient Rule provides a way to find the derivative without rearranging the terms unnecessarily.To apply the Quotient Rule, remember the structure of this rule: if you have two functions, \( u(x) \) and \( v(x) \), the derivative of their quotient is given by:

• \( \left(\frac{f}{g}\right)' = \frac{gf' - fg'}{g^2} \)

Here, \( f \) is the numerator function and \( g \) is the denominator function. In this specific exercise, you'd set \( f = 4-x^3 \) and \( g = x-x^2 \). Calculating the derivatives of \( f \) and \( g \) first (\( f' = -3x^2 \) and \( g' = 1-2x \)) simplifies the process of differentiating more complex functions.The purpose of utilizing the Quotient Rule here is to systematically handle the complexity that arises from having a numerator and denominator, ensuring each part of the expression is appropriately accounted for in the derivative.

Chain Rule

The Chain Rule is an essential concept in calculus for differentiating composite functions, where one function nests inside another. In this exercise, there's a cube root applied to a quotient, meaning the Chain Rule simplifies the process of differentiation.The Chain Rule essentially states:

• \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \)

This means you first differentiate the outer function while keeping the inner function unchanged, and then multiply by the derivative of the inner function.For the given function \( f(x) = \sqrt[3]{\frac{4-x^3}{x-x^2}} \), think of it as \( (u)^{1/3} \) where \( u = \frac{4-x^3}{x-x^2} \). Apply the Chain Rule by first differentiating the outer function \( (u)^{1/3} \) to get \( \frac{1}{3} u^{-2/3} \), and then multiply by the derivative of the inside, \( u'(x) \), which we find using the Quotient Rule. This layered approach simplifies handling functions within functions, making complex derivatives more manageable.

Simplifying Expressions

Once derivatives are taken, especially with complex functions such as the one in our exercise, simplification becomes an essential step. Simplifying expressions is crucial to make them more understandable and user-friendly.In this scenario, even after you've applied both the Quotient Rule and the Chain Rule, it's very likely the expression will be complicated. Consider the expanded terms: \( -3x^3 + 3x^4 \) from \((x-x^2)(-3x^2)\) part, and \( 4 - 8x - x^3 + 2x^4 \) from \((4-x^3)(1-2x)\). The term combinations might initially seem daunting.To simplify effectively:

• Combine like terms: look for terms you can add or subtract directly.
• Factor where possible, to reduce polynomial degree or cancel terms.
• Rewrite fractions using common denominators.

Simplifying helps to reveal the true nature of the derivative function, making analysis and application easier in both homework and real-world contexts. It reduces the risk of errors that can stem from dealing with overly cumbersome expressions.