Problem 37 - Calculus an Applied Approach

Question

Find \(f^{\prime}(x)\) $$ f(x)=x^{2}-2 x-\frac{2}{x^{4}} $$

Step-by-Step Solution

Answer

The derivative \(f'(x)\) is \(2x - 2 + 8x^{-5}\).

Step 1: Differentiate \(x^2\)

The derivative of \(x^{2}\) is given by bringing the exponent in front and reducing the exponent by 1. Therefore, the derivative of \(x^{2}\) is \(2x\).

Step 2: Differentiate \(-2x\)

The derivative of \(2x\) is \(-2\) because the derivative of any constant multiplied by \(x\) is simply the constant.

Step 3: Differentiate \(-2/x^{4}\)

First, rewrite \(-2/x^{4}\) as \(-2x^{-4}\). The derivative of this can be calculated by bringing the exponent in front and reducing the exponent by 1, giving us \(8x^{-5}\).

Step 4: Combine the Results

Adding up the results from the above steps, we find that the derivative \(f'(x) = 2x - 2 + 8x^{-5}\).

Key Concepts

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. They can have one or more terms, such as constant terms, linear terms, quadratic terms, and so forth. Expressing functions clearly can help in understanding how derivatives are worked out.

• In our exercise, the term \(x^2\) is a part of the polynomial, where the highest power of the variable is 2, making it a quadratic function.
• Another term, \(-2x\), is a linear term since the variable is raised to the power of 1.
• Polynomial functions are familiar and widely used because they are smooth and continuous, making them easy to work with in calculus.

When we say a polynomial function is in a simplified form, it means all like terms have been combined, and it is in its simplest arrangement.

Power Rule

The power rule is a basic differentiation tool that is easy to apply to find the derivative of most polynomial terms. It states that if you have a term \(ax^n\), its derivative is \(n \cdot ax^{n-1}\). This powerful rule makes it simple to differentiate not just one term, but each term of any polynomial separately.

• As shown in the exercise, using the power rule to differentiate \(x^2\) gives us \((2)x^{2-1} = 2x\).
• For a constant multiplied term like \(-2x\), applying the rule directly provides \(n(-2)x^{1-1} = -2\), because the derivative of \(x\) is 1.
• Sometimes, the term might not initially appear in a suitable form for the power rule, such as in \(-2/x^4\). Rewriting it as \(-2x^{-4}\) allows us to apply the power rule effectively. The new derivative becomes \( (-4)(-2)x^{-4-1} = 8x^{-5}\).

Understanding and mastering the power rule helps streamline calculations, especially when dealing with polynomials.

Derivative of Rational Functions

Rational functions are ratios of polynomial functions, expressed as \(f(x) = \frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials. Finding derivatives of rational functions involves different rules.

• When a function includes a term like \(\frac{2}{x^4}\), it might be helpful to rewrite it as \(-2x^{-4}\). This converts a rational function into a polynomial form which is easier to differentiate.
• Instead of using the quotient rule, which is another method for derivatives, recognizing and transforming the term helps in directly applying the power rule—a faster and often simpler approach.

The process of "rewriting" is a valuable skill that simplifies differentiation and makes handling complex expressions straightforward. In many cases, dealing with rational functions becomes more about simplifying terms first, essentially turning them into polynomial-like structures for ease of differentiation.