Problem 52 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{10x^{5} - 6x^{4} - 4x^{3} + 26x^{2}}{(5x^{2} - 2x + 1)^{2}} \).

Step 1: Identify the Differentiation Rule

Given the function \( f(x) = \frac{x^{4} + 2x - 1}{5x^{2} - 2x + 1} \), we'll use the quotient rule for differentiation, which is: \( \left( \frac{u}{v} \right)' = \frac{vu' - uv'}{v^{2}} \). Here, let \( u = x^{4} + 2x - 1 \) and \( v = 5x^{2} - 2x + 1 \).

Step 2: Differentiate the Numerator

Differentiate \( u = x^{4} + 2x - 1 \) with respect to \( x \). The derivative \( u' \) is derived as follows: - The derivative of \( x^{4} \) is \( 4x^{3} \).- The derivative of \( 2x \) is \( 2 \).- The derivative of the constant \(-1\) is \( 0 \).Therefore, \( u' = 4x^{3} + 2 \).

Step 3: Differentiate the Denominator

Differentiate \( v = 5x^{2} - 2x + 1 \) with respect to \( x \). The derivative \( v' \) is obtained:- The derivative of \( 5x^{2} \) is \( 10x \).- The derivative of \(-2x\) is \(-2\).- The derivative of \(1\) is \(0\).Thus, \( v' = 10x - 2 \).

Step 4: Apply the Quotient Rule

Now apply the quotient rule: \[ f'(x) = \frac{(5x^{2} - 2x + 1)(4x^{3} + 2) - (x^{4} + 2x - 1)(10x - 2)}{(5x^{2} - 2x + 1)^{2}} \]Simplify the expression in the numerator by expanding both products and combining like terms.

Step 5: Simplify the Numerator

Expand each product in the numerator:\((5x^{2} - 2x + 1)(4x^{3} + 2) = 20x^{5} + 10x^{2} - 8x^{4} - 4x^{2} + 4x^{3} + 2\).\((x^{4} + 2x - 1)(10x - 2) = 10x^{5} - 20x^{2} + 8x^{3} - 2x^{4} - 2\).Combine and simplify:\(20x^{5} + 10x^{2} - 8x^{4} - 4x^{2} + 4x^{3} + 2 - (10x^{5} - 20x^{2} + 8x^{3} - 2x^{4} - 2)\).

Step 6: Combine and Simplify Terms

Combine all terms from the expanded numerator:\[ 20x^{5} - 10x^{5} - 8x^{4} + 2x^{4} + 4x^{3} - 8x^{3} + 10x^{2} - 4x^{2} + 20x^{2} - 2 + 2 \].Simplify to get:\[ 10x^{5} - 6x^{4} - 4x^{3} + 26x^{2} \].Thus, the derivative is:\[ f'(x) = \frac{10x^{5} - 6x^{4} - 4x^{3} + 26x^{2}}{(5x^{2} - 2x + 1)^{2}} \].

Key Concepts

Quotient Rule

When dealing with functions expressed as one polynomial divided by another, namely \( \frac{u}{v} \), we use the quotient rule to find their derivatives. If you find a fraction in a function, that's your clue that the quotient rule will be handy. This rule states that the derivative is \( \left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^{2}} \). This means:

• First, find the derivative of the numerator \( u' \)
• Then find the derivative of the denominator \( v' \)
• Plug these into the formula, subtracting the product of the original numerator \( u \) and \( v' \) from the product of the original denominator \( v \) and \( u' \)
• Divide the whole expression by the square of the original denominator \( v^2 \)

Using the quotient rule efficiently can greatly simplify solving complex rational functions by systematically dealing with each component.

Derivatives

Derivatives are a fundamental concept in calculus representing the rate of change of one quantity with respect to another. When we say we are differentiating \( f(x) \), we mean we are finding its derivative, often denoted as \( f'(x) \). This process involves applying rules like the power rule, product rule, and quotient rule depending on the form of the function. For example, the derivative of \( x^n \) is \( nx^{n-1} \), which is known as the power rule.In the given problem, when differentiating the numerator \( x^4 + 2x - 1 \) and the denominator \( 5x^2 - 2x + 1 \), we used this principle to determine:

• The derivative of \( x^4 \) is \( 4x^3 \)
• The derivative of \( 2x \) is \( 2 \)
• The derivative of any constant, such as \( -1 \), is \( 0 \)

Similarly, these rules helped differentiate the denominator efficiently. Understanding how to apply these rules makes it easier to handle derivatives and is key to successfully using the quotient rule.

Function Simplification

Simplifying a function, particularly after applying differentiation rules, is key to making the function manageable and easier to interpret. It involves combining like terms and reducing the expression to the simplest form.After applying the quotient rule in our problem, we expanded and combined terms in the numerator to simplify the expression. Here are steps to guide through the simplification:

• Expand each product individually in the numerator.
• Combine like terms by collecting similar powers of \( x \).
• Ensure all unnecessary terms are canceled out or simplified.

This systematic approach reduces errors and results in a clean and clear expression, as seen in the transformation from a complex set of terms to \( 10x^5 - 6x^4 - 4x^3 + 26x^2 \). Simplification is often the final step in differentiation, leaving you with a neat derivative form that is ready for further evaluation or application.