Problem 50 - Calculus for Biology and Medicine

Question

Differentiate with respect to the independent variable. $$ f(x)=\frac{1-4 x^{3}}{1-x} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{8x^3 - 12x^2 + 1}{(1-x)^2} \).

Step 1: Identify the Form of the Function

The function given is a rational function of the form \( f(x) = \frac{g(x)}{h(x)} \), where \( g(x) = 1 - 4x^3 \) and \( h(x) = 1 - x \). To differentiate this, we will use the quotient rule for differentiation.

Step 2: Recall the Quotient Rule

The quotient rule states that if you have a function \( \frac{u(x)}{v(x)} \), its derivative is \( \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \). For our function, set \( u(x) = 1 - 4x^3 \) and \( v(x) = 1 - x \).

Step 3: Differentiate the Numerator

Differentiate \( u(x) = 1 - 4x^3 \) with respect to \( x \). The derivative is \( u'(x) = -12x^2 \).

Step 4: Differentiate the Denominator

Differentiate \( v(x) = 1 - x \) with respect to \( x \). The derivative is \( v'(x) = -1 \).

Step 5: Apply the Quotient Rule

Using the quotient rule, the derivative \( f'(x) \) is given by: \[ f'(x) = \frac{(-12x^2)(1-x) - (1-4x^3)(-1)}{(1-x)^2} \].

Step 6: Simplify the Derivative Expression

Simplify the expression obtained in Step 5. Expand both products: \( -12x^2(1-x) = -12x^2 + 12x^3 \) and \((1-4x^3)(-1) = -1 + 4x^3 \). Thus, the numerator becomes:\[-12x^2 + 12x^3 + 1 - 4x^3 \]which simplifies to \[8x^3 - 12x^2 + 1.\] So, \[ f'(x) = \frac{8x^3 - 12x^2 + 1}{(1-x)^2}. \]

Step 7: Present the Final Simplified Derivative

The final expression for the derivative of the function \( f(x) = \frac{1-4 x^{3}}{1-x} \) is:\[ f'(x) = \frac{8x^3 - 12x^2 + 1}{(1-x)^2}. \]

Key Concepts

Quotient Rule

One of the cornerstones of differentiating rational functions is the quotient rule. When you need to derive a function of the form \( \frac{u(x)}{v(x)} \), remember the quotient rule formula: \[ \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \]. This formula helps simplify the process of finding the derivative of a function that involves a division of two different expressions. It is crucial because direct differentiation of complex expressions may be cumbersome or impractical.
To use the quotient rule, follow these simple steps:

• Identify the functions \( u(x) \) and \( v(x) \) from the numerator and denominator respectively.
• Find \( u'(x) \) by differentiating \( u(x) \).
• Find \( v'(x) \) by differentiating \( v(x) \).
• Apply the formula to get the derivative \( \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \).

By memorizing and understanding the quotient rule, you will be more equipped to handle problems involving divisions of differentiable functions.

Rational Functions

Rational functions are an essential part of calculus incorporating both polynomial functions and their ratios. A rational function is expressed as the quotient of two polynomial functions and is written in the form \( f(x) = \frac{g(x)}{h(x)} \).
Understanding the characteristics of rational functions is vital for differentiating, integrating, and analyzing them effectively. A few key points about rational functions include:

• They can display horizontal and vertical asymptotes due to the division nature, often leading to undefined points where the denominator is zero.
• They can model both real-world and abstract mathematical concepts, like rates of growth and decay.
• Simplification before differentiation can sometimes ease solving if common factors exist in the numerator and denominator.

Grasping how to manipulate and differentiate these functions using rules like the quotient rule will enhance your problem-solving capabilities in calculus.

Calculus Problem-Solving

Successfully solving calculus problems often requires a consistent approach along with a firm grasp of fundamental concepts like the quotient rule, rational functions, and derivatives. When faced with a problem that requires differentiation, take these steps to strategically approach and solve it:

• Identify the Form of the Function: Recognize if the function fits any well-known forms, such as rational or polynomial. This helps in choosing the correct rule or method, like the quotient rule for division-based expressions.
• Simplify Where Possible: If the function or its expression can be simplified before applying derivative rules, it often saves time and reduces mistakes.
• Apply Appropriate Rules Sequentially: Use differentiation rules like the power rule, product rule, and quotient rule where necessary, one at a time, ensuring each step is handled correctly.
• Simplify the Expression: After finding the derivative, always simplify the resultant expression for a cleaner and more comprehensible result.

By following these methodical steps, calculus problem-solving becomes much more manageable, allowing you to tackle even the more complex differentiation problems with confidence.