Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of each function. \(f(x)=\left(x^{2}-5 x+2\right)\left(x-\frac{2}{x}\right)\)

Step-by-Step Solution

Answer

The derivative of the given function is \(f'(x) = 3x^2 - 10x - 2 + \frac{2}{x}\).

Step 1: Identify the functions g(x) and h(x)

In this problem, we have the function \(f(x) = (x^2 - 5x + 2)(x - \frac{2}{x})\). We can identify the functions g(x) and h(x) as follows: \(g(x) = x^2 - 5x + 2\) \(h(x) = x - \frac{2}{x}\)

Step 2: Find the derivative of g(x)

We will first find the derivative of \(g(x) = x^2 - 5x + 2\). Using basic rules of differentiation, we have: \(g'(x) = 2x - 5\)

Step 3: Find the derivative of h(x)

Next, we will find the derivative of \(h(x) = x - \frac{2}{x}\). We rewrite the function as a sum of powers of x and then use basic differentiation rules: \(h(x) = x - 2x^{-1}\) \(h'(x) = 1 + 2x^{-2}\)

Step 4: Apply the Product Rule

Now, we will apply the product rule to find the derivative of the given function: \(f'(x) = g'(x)h(x) + g(x)h'(x)\) Substitute the derivatives and functions we found in the previous steps: \(f'(x) = (2x - 5)(x - \frac{2}{x}) + (x^2 - 5x + 2)(1 + 2x^{-2})\)

Step 5: Simplify the expression

Finally, we will simplify the given expression by multiplying and combining like terms: \(f'(x) = 2x^2 - 5x - 4 + \frac{10}{x} + x^2 - 5x + 2 + 2x^{-1} - 10x^{-1}\) \(f'(x) = 3x^2 - 10x - 2 + \frac{2}{x}\) This is the derivative of the given function.

Key Concepts

Product Rule

The product rule is a crucial technique in calculus for finding the derivative of a product of two functions. When you have two functions, say \( g(x) \) and \( h(x) \), the product rule allows you to determine the derivative of their product \( f(x) = g(x)h(x) \).

Here's how it works:

• First, find the derivative of \( g(x) \), which we denote as \( g'(x) \).
• Next, find the derivative of \( h(x) \), denoted as \( h'(x) \).
• Then, apply the product rule formula: \( f'(x) = g'(x)h(x) + g(x)h'(x) \).

This formula essentially states that the derivative of a product is the sum of each function multiplied by the derivative of the other. It's a handy rule because it simplifies what could be a tedious process, and it ensures accuracy in solving problems involving product functions.

Mastering the product rule can greatly enhance your ability to tackle more complex calculus problems.

Power Rule

The power rule is one of the simplest yet most powerful tools in differentiation, allowing you to find derivatives of powers of \( x \) very efficiently. If you have a function of the form \( x^n \), the power rule states that its derivative is \( nx^{n-1} \).

This rule is straightforward:

• Take the exponent \( n \) and multiply it by the base \( x \) raised to the power of \( n-1 \).

For example, if you have \( g(x) = x^2 \), applying the power rule gives you \( g'(x) = 2x^{2-1} = 2x \).

As seen in the exercise, using the power rule on \( g(x) = x^2 - 5x + 2 \) easily gives the derivative \( g'(x) = 2x - 5 \).

This rule simplifies the derivative process, making it easier to handle polynomial functions quickly.

Derivative Computation

Computing derivatives requires understanding how different rules like the power rule and the product rule work together in functions. Derivatives tell us the rate at which a function is changing at any given point.

In the exercise, you compute the derivative of a composite function \( f(x) = (x^2 - 5x + 2)(x - \frac{2}{x}) \) by first differentiating the component parts, \( g(x) \) and \( h(x) \). Then, apply the derivative rules to find \( g'(x) \) and \( h'(x) \).

By solving these separately:

• \( g'(x) = 2x - 5 \)
• \( h'(x) = 1 + 2x^{-2} \)

You can then combine them using the product rule to compute the overall derivative \( f'(x) \).

This structured approach makes it easier to handle complex functions by breaking them down into simpler pieces.

Simplification of Expressions

Once you have computed the derivative of a function, it is often beneficial to simplify the expression. Simplification helps in understanding the behavior of the function better and makes it easier to analyze or visualize.

For example, after applying the product rule, the derivative might appear complex and messy. In the exercise, the derivative was initially obtained as:

\[ f'(x) = 2x^2 - 5x - 4 + \frac{10}{x} + x^2 - 5x + 2 + 2x^{-1} - 10x^{-1} \]

To simplify:

• Combine like terms from all similar powers of \( x \).
• Reassess any fractions for common denominators.

This leads to a clean and concise form: \( f'(x) = 3x^2 - 10x - 2 + \frac{2}{x} \).

Simplification is a rewarding step that clarifies the derivative and often makes interpretations or subsequent calculations more straightforward.