Problem 54 - Calculus for Biology and Medicine

Question

Differentiate with respect to the independent variable. \(f(x)=\frac{1+2 x^{2}-4 x^{4}}{3 x^{3}-5 x^{5}}\)

Step-by-Step Solution

Answer

The derivative of the function is obtained using the quotient rule; exact expression requires further algebraic simplification.

Step 1: Identify Function Structure

The function given is a rational function: \[f(x) = \frac{1 + 2x^2 - 4x^4}{3x^3 - 5x^5}\]The numerator is \(1 + 2x^2 - 4x^4\) and the denominator is \(3x^3 - 5x^5\). We need to differentiate this function using the quotient rule.

Step 2: Recall the Quotient Rule

The quotient rule for differentiation of a function \(\frac{u}{v}\) is given by:\[\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}\]Here, \(u = 1 + 2x^2 - 4x^4\) and \(v = 3x^3 - 5x^5\).

Step 3: Differentiate the Numerator

The numerator is \(u = 1 + 2x^2 - 4x^4\). Differentiating each term with respect to \(x\), we get:\[\frac{du}{dx} = 0 + 4x - 16x^3 = 4x - 16x^3\]

Step 4: Differentiate the Denominator

The denominator is \(v = 3x^3 - 5x^5\). Differentiating each term with respect to \(x\), we find:\[\frac{dv}{dx} = 9x^2 - 25x^4\]

Step 5: Apply the Quotient Rule

Using the quotient rule:\[\frac{d}{dx}\left(\frac{1 + 2x^2 - 4x^4}{3x^3 - 5x^5}\right) = \frac{(3x^3 - 5x^5)(4x - 16x^3) - (1 + 2x^2 - 4x^4)(9x^2 - 25x^4)}{(3x^3 - 5x^5)^2}\]

Step 6: Simplify the Expression

Calculate each part of the expression obtained in step 5. First, simplify the expression in the numerator:- Expand and simplify \((3x^3 - 5x^5)(4x - 16x^3)\).- Expand and simplify \((1 + 2x^2 - 4x^4)(9x^2 - 25x^4)\).- Subtract the latter from the former to complete the numerator.- Denominator remains \((3x^3 - 5x^5)^2\).This step involves performing algebraic multiplication and simplifications, which are straightforward but lengthy steps.

Key Concepts

Understanding the Quotient Rule

The quotient rule is crucial for differentiating rational functions, which are ratios of two polynomials. When you have a function organized as a division, specifically \(\frac{u}{v}\), the quotient rule helps you find the derivative. It states:

• Differentiate the numerator (\(u\)), get \(\frac{du}{dx}\)
• Differentiate the denominator (\(v\)), to find \(\frac{dv}{dx}\)
• Apply the quotient rule formula:\[\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}\]
• Simplify, if possible, to get the cleanest form of the derivative.

The quotient rule helps manage the complexities that arise from differentiating divisions of functions. This approach prevents the errors of differentiating each function part separately and then dividing, which would not yield the correct derivative.

Exploring Rational Functions

Rational functions are formed when one polynomial is divided by another. These functions look like this: \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomial expressions. A good example is given by the problem: \(f(x) = \frac{1 + 2x^2 - 4x^4}{3x^3 - 5x^5}\).

• The numerator (top part) is \(1 + 2x^2 - 4x^4\)
• The denominator (bottom part) is \(3x^3 - 5x^5\)

These functions are significant in calculus because they are common in real-world applications where quantities are naturally related by division. Rational functions may have undefined points, discontinuities, or asymptotes based on the values of \(x\) that make the denominator zero. Understanding these functions requires both their algebraic manipulation and graphical representation.

Digging into Derivatives

Derivatives form the backbone of calculus by describing how a function changes. For any given function, the derivative tells us the rate of change or the slope of the function at any point. This is essential for understanding motion, growth, and trends.
To differentiate the rational function \(f(x) = \frac{1 + 2x^2 - 4x^4}{3x^3 - 5x^5}\), we apply the quotient rule:

• Differentiate the numerator: \(\frac{du}{dx} = 4x - 16x^3\)
• Differentiate the denominator: \(\frac{dv}{dx} = 9x^2 - 25x^4\)
• Substitute these derivatives into the quotient rule formula

The differentiation simplifies the insights into the behavior of the function, revealing where it increases or decreases, and identifies critical points, inflections, and asymptotes.