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

Apply the Quotient Rule: \(f'(x) = \frac{20x^5 - 8x^4 + 4x^3 + 10x^2 - 4x + 2}{(5x^2 - 2x + 1)^2}\).

Step 1: Identify the Quotient Rule

The function is a quotient, so we use the Quotient Rule. If you have a function \(f(x) = \frac{u(x)}{v(x)}\), the derivative is \(f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}\). Here \(u(x) = x^4 + 2x - 1\) and \(v(x) = 5x^2 - 2x + 1\).

Step 2: Differentiate the Numerator

Find \(u'(x)\), the derivative of the numerator. Given \(u(x) = x^4 + 2x - 1\), differentiate each term: \(u'(x) = 4x^3 + 2\).

Step 3: Differentiate the Denominator

Find \(v'(x)\), the derivative of the denominator. Given \(v(x) = 5x^2 - 2x + 1\), differentiate each term: \(v'(x) = 10x - 2\).

Step 4: Apply the Quotient Rule

Substitute the derivatives into the Quotient Rule: \(f'(x) = \frac{(4x^3 + 2)(5x^2 - 2x + 1) - (x^4 + 2x - 1)(10x - 2)}{(5x^2 - 2x + 1)^2}\).

Step 5: Simplify the Expression

Simplify the expression for \(f'(x)\):- Expand \((4x^3 + 2)(5x^2 - 2x + 1)\) and \((x^4 + 2x - 1)(10x - 2)\).- Simplify the numerator by combining like terms.- Leave the simplified numerator expression over \((5x^2 - 2x + 1)^2\).

Step 6: Present the Final Derivative Expression

After simplification, we obtain the final derivative:\[ f'(x) = \frac{20x^5 - 8x^4 + 4x^3 + 10x^2 - 4x + 2}{(5x^2 - 2x + 1)^2} \].

Key Concepts

Derivative

The derivative of a function is a fundamental concept in calculus, measuring how the function's output changes as its input changes. In simpler terms, it can be thought of as the rate at which something happens. For example, if you are given a curve representing a car's journey, the derivative would tell you the car's speed at any given moment.

In our case, the function given is in the form of a quotient, specifically a fraction where one polynomial is divided by another. This means to find the derivative, we will use the Quotient Rule. It's important to understand that the derivative of a fraction like this is not simply the derivative of the top divided by the derivative of the bottom. Instead, you need to apply a special formula which involves the derivatives of both the numerator and the denominator.

This brings us to the key formula of the Quotient Rule: if your function is represented as \( f(x) = \frac{u(x)}{v(x)} \), then its derivative is given by \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \). This formula will help us determine how both parts of the fraction contribute to the overall change in \( f(x) \).

Numerator and Denominator Differentiation

Differentiating the numerator and denominator separately is a crucial step when applying the Quotient Rule. Each part involves finding the derivative of a polynomial expression.

Let's start with the numerator, \( u(x) = x^4 + 2x - 1 \). When differentiating, treat each term independently:

• The derivative of \( x^4 \) is \( 4x^3 \).
• The derivative of \( 2x \) is simply 2.
• The derivative of a constant, -1, is 0.

Thus, the derivative of the numerator, \( u'(x) \), is \( 4x^3 + 2 \).

Next, focus on the denominator, \( v(x) = 5x^2 - 2x + 1 \). Differentiate each term:

• \( 5x^2 \) becomes \( 10x \).
• \( -2x \) becomes -2.
• As always, the derivative of a constant, 1, is 0.

So, the derivative of the denominator, \( v'(x) \), is \( 10x - 2 \). With both derivatives calculated, you're ready to substitute them into the Quotient Rule formula.

Simplifying Expressions

Once you've substituted the differentiated parts into the Quotient Rule, the result may seem quite complicated initially. But simplifying the expression is a rewarding process that clarifies the final answer.

Upon substitution, the numerator becomes a large expression:

• \((4x^3 + 2)(5x^2 - 2x + 1)\)
• Minus \((x^4 + 2x - 1)(10x - 2)\).

This complexity often results from the derivatives' multiplication and involves combining like terms. Expansion and simplification require patience, as you distribute each term in parentheses across the others. For clarity:

• Expand both sets of terms fully, multiplying each part.
• Combine similar terms, such as collecting all the \(x^5\) terms together, then the \(x^4\) terms, and so forth.

After simplification of the numerator via careful arithmetic and gathering alike powers, you'll arrive at a single fraction expressing \( f'(x) \), with \( (5x^2 - 2x + 1)^2 \) remaining in the denominator. Finally, you will have a neatly simplified derivative: \[ f'(x) = \frac{20x^5 - 8x^4 + 4x^3 + 10x^2 - 4x + 2}{(5x^2 - 2x + 1)^2}. \] This cleaner expression is much easier to interpret and use in further analysis or application.