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

Step 1: Identify the Function Type

The function given is a rational function, where the numerator and denominator are polynomials: \(f(x) = \frac{1 - 4x^3}{1 - x}\). We will need to use the quotient rule to differentiate this rational function.

Step 2: Recall the Quotient Rule

The quotient rule for differentiation states that if you have a function \(f(x) = \frac{g(x)}{h(x)}\), then its derivative \(f'(x)\) is given by \(f'(x) = \frac{g'(x) h(x) - g(x) h'(x)}{(h(x))^2}\). Here, \(g(x) = 1 - 4x^3\) and \(h(x) = 1 - x\).

Step 3: Differentiate the Numerator and Denominator

Differentiate \(g(x) = 1 - 4x^3\):- \(g'(x) = 0 - 12x^2 = -12x^2\).Differentiate \(h(x) = 1 - x\):- \(h'(x) = 0 - 1 = -1\).

Step 4: Apply the Quotient Rule

Substitute the derivatives and original functions into the quotient rule formula:\[ f'(x) = \frac{(-12x^2)(1 - x) - (1 - 4x^3)(-1)}{(1 - x)^2}\]Now simplify this expression.

Step 5: Simplify the Expression

Simplify the numerator step by step:\[\begin{aligned}& (-12x^2)(1 - x) = -12x^2 + 12x^3, \& (1 - 4x^3)(-1) = -1 + 4x^3.\end{aligned}\]Combine these results:\[f'(x) = \frac{-12x^2 + 12x^3 + 1 - 4x^3}{(1 - x)^2}\]Next, combine like terms:\[f'(x) = \frac{-12x^2 + 8x^3 + 1}{(1 - x)^2}\].

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, focusing on the rate at which things change. It's about finding a function's derivative, which is like the speed of change at any given point. By differentiating, we gain insight into the slope of a function at any point on its curve. This crucial tool helps solve real-world problems involving rates of change, such as velocity in physics or growth rates in biology. For a function represented as \(f(x)\), its derivative, noted as \(f'(x)\), tells us how \(f(x)\) changes as \(x\) changes.

Let's break down the differentiation process using the function \(f(x) = \frac{1 - 4x^3}{1 - x}\). This function is a rational function, meaning it needs the quotient rule. Remember that the quotient rule is vital for differentiating when you have two functions divided by each other:

• We identify \(g(x) = 1 - 4x^3\)
• and \(h(x) = 1 - x\).
• Then differentiate each to get \(g'(x) =-12x^2\) and \(h'(x) = -1\).

Using these derivatives and the quotient rule, we calculate \(f'(x)\), and by following a sequence of steps, find the derivative.

Rational Function

A rational function is a type of function represented as the ratio of two polynomials. Think of a fraction where the top (numerator) and bottom (denominator) are polynomial expressions. The rational function form is \(f(x) = \frac{g(x)}{h(x)}\), where both \(g(x)\) and \(h(x)\) are polynomials.

In our example, \(f(x) = \frac{1 - 4x^3}{1 - x}\), you see clearly that the function is rational since both the numerator, \(1 - 4x^3\), and the denominator, \(1 - x\), are polynomials. Rational functions have unique characteristics:

• They can have vertical asymptotes, which occur where the denominator is zero (meaning undefined).
• Simplifying them often helps gain insight into their behavior.
• Problems involving rational functions frequently require tools such as the quotient rule for differentiation.

Understanding how to work with rational functions is essential in calculus, especially when tasked with finding derivatives.

Polynomial Functions

Polynomial functions are the building blocks of many complex functions, recognizable by their form: \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where each \(a_i\) represents a constant coefficient, and the highest power \(n\) is a non-negative integer. Each term is a product of a coefficient and a variable raised to an exponent.

In the differentiation exercise, you encounter polynomial functions in both the numerator \(1 - 4x^3\) and the denominator \(1 - x\). Here, these functions dictate the behavior of their rational counterpart. Understanding the properties of polynomial functions helps immensely:

• Easy to differentiate because the power rule applies: \(\frac{d}{dx}[x^n] = nx^{n-1}\).
• Simple to manipulate and combine, making them foundational in calculus.
• Understanding how they behave, with respect to slope and curvature, is crucial for analyzing the rational functions they form.

As you deal with complex expressions, identifying polynomial parts can make solving calculus problems far more manageable.