Problem 4 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Find the derivative of the given function. $$ f(x)=x^{3}-\frac{1}{3 x^{5}}+2 \sqrt{x}-\frac{3}{x}+\frac{1-2 x}{x^{3}} $$

Step-by-Step Solution

Answer

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

Step 1 - Rewrite the Function

Rewrite the given function in a more convenient form for differentiation:\[ f(x) = x^3 - \frac{1}{3}x^{-5} + 2x^{1/2} - 3x^{-1} + \frac{1}{x^3} - \frac{2x}{x^3} \]Simplify the last term:\[ f(x) = x^3 - \frac{1}{3}x^{-5} + 2x^{1/2} - 3x^{-1} + x^{-3} - 2x^{-2} \]

Step 2 - Apply the Power Rule

Differentiate each term using the power rule, which states that \( \frac{d}{dx} [x^n] = nx^{n-1} \):\[ \frac{d}{dx}\big( x^3 \big) = 3x^2 \]\[ \frac{d}{dx} \big( -\frac{1}{3} x^{-5} \big) = -\frac{1}{3} (-5)x^{-6} = \frac{5}{3}x^{-6} \]\[ \frac{d}{dx} \big( 2x^{1/2} \big) = 2 \cdot \frac{1}{2} x^{-1/2} = x^{-1/2} \]\[ \frac{d}{dx} \big( -3x^{-1} \big) = -3(-1)x^{-2} = 3x^{-2} \]\[ \frac{d}{dx} \big( x^{-3} \big) = -3x^{-4} \]\[ \frac{d}{dx} \big( -2x^{-2} \big) = -2(-2)x^{-3} = 4x^{-3} \]

Step 3 - Combine the Results

Combine all the differentiated terms to obtain the derivative of the function:\[ f'(x) = 3x^2 + \frac{5}{3}x^{-6} + x^{-1/2} + 3x^{-2} - 3x^{-4} + 4x^{-3} \]

Key Concepts

Power Rule

The power rule is a fundamental technique in calculus used to find the derivative of functions involving powers of \(x\). It's straightforward and very handy for polynomial functions. The power rule states that if you have a term \(x^n\), the derivative is \(nx^{n-1}\). It’s a simple but powerful rule.
For example:
\[ \frac{d}{dx} \big( x^3 \big) = 3x^2 \]
Note that we multiply the exponent by the base and then reduce the exponent by 1. This technique is applied to each term in a polynomial or other power terms, making differentiation much simpler. Let's look at how this works in the original problem:
We had terms like \(x^3, x^{-5}, x^{1/2}\) and more. Applying the power rule makes it easier to handle each term individually.

Derivative

The concept of the derivative represents the rate of change of a function with respect to a variable. In simpler terms, it shows how a function changes as its input changes. Derivatives are a core concept in calculus and have practical applications in science, engineering, and economics.
To find the derivative, especially for functions like polynomials, we use rules like the power rule: \( \frac{d}{dx}[x^n] = nx^{n-1} \). This rule was applied step-by-step in the original problem for each term in the function. For instance:
\[ \frac{d}{dx} \big( -\frac{1}{3} x^{-5} \big) = \frac{5}{3}x^{-6} \]
We multiplied the exponent (-5) by the coefficient (-1/3) and then decreased the exponent by 1, resulting in \(5/3x^{-6}\). Each term in the function undergoes similar treatment to find the overall derivative.

Simplification

Simplification in calculus involves making expressions easier to work with by reducing complexity. It's a crucial step before applying differentiation. In this exercise, we first rewrote the function in a more straightforward form before applying the power rule.
For instance, terms like \(\frac{1-2x}{x^3}\) were simplified:
\[ f(x) = x^3 - \frac{1}{3} x^{-5} + 2x^{1/2} - 3x^{-1} + x^{-3} - 2x^{-2} \]
Breaking down complex terms into simpler components can make differentiation much more manageable. This approach helped when applying the power rule to get each term’s derivative individually. Simplification not only makes differentiation easier but also helps in combining like terms efficiently.