Problem 16 - Brief Applied Calculus

Question

Find the derivative of each function. $$ f(x)=\frac{1}{x^{2 / 3}} $$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Function Using Exponents

First, rewrite the function \( f(x) = \frac{1}{x^{2/3}} \) as an expression with a negative exponent. This is done because differentiation rules for power functions are easier to apply:\[ f(x) = x^{-2/3} \]

Step 2: Apply the Power Rule for Differentiation

The derivative of \( f(x) = x^n \) with respect to \( x \) is \( f'(x) = n x^{n-1} \). Apply this power rule to \( f(x) = x^{-2/3} \):- Identify \( n = -\frac{2}{3} \).- Differentiate to find \( f'(x) \):\[ f'(x) = -\frac{2}{3} x^{(-2/3)-1} \]

Step 3: Simplify the Expression

Simplify the expression for the derivative:- Calculate the new exponent: \(-\frac{2}{3} - 1 = -\frac{2}{3} - \frac{3}{3} = -\frac{5}{3}\).Therefore, the derivative is:\[ f'(x) = -\frac{2}{3} x^{-5/3} \]

Step 4: Rewrite Using Positive Exponents (Optional)

Although not always necessary, rewriting the derivative with positive exponents can sometimes be useful for interpretation:\[ f'(x) = -\frac{2}{3} \frac{1}{x^{5/3}} \]

Key Concepts

Power Rule in Differentiation

The Power Rule is a fundamental tool in calculus for finding derivatives. It's especially convenient when dealing with polynomial functions. When you have a function in the form of \( f(x) = x^n \), the Power Rule states that the derivative \( f'(x) \) is \( n \cdot x^{n-1} \). Easy to remember and apply, this rule is crucial for tackling more complex calculus problems.
You apply the Power Rule when differentiating a variety of functions:

• Normal powers like \( x^2, x^3, \) etc.
• Fractional powers such as \( x^{1/2}, x^{-3/4} \).
• Even negative powers or exponents involved.

In the example \( f(x) = x^{-2/3} \), you identified \( n = -2/3 \) and calculated the derivative as \( -\frac{2}{3} x^{-5/3} \), applying the rule that simplifies differentiation for exponential expressions.

Understanding Negative Exponents

Negative exponents might seem tricky at first, but they're quite simple after you understand their meaning. A negative exponent indicates reciprocation. That is, \( a^{-n} = \frac{1}{a^n} \). This simple translation helps when differentiating or manipulating expressions with exponents.
This concept is used in the exercise by rewriting the function \( f(x) = \frac{1}{x^{2/3}} \) as \( x^{-2/3} \). The conversion from a fraction to a negative exponent is crucial for applying the Power Rule in differentiation.
Here's when to think about negative exponents in calculus:

• When rewriting fractions to facilitate differentiation.
• When simplifying expressions for integration or differentiation.

Always remember, a negative exponent makes differentiation more seamless and aligns your problem with standard calculus rules.

Simplification in Calculus

Simplification is a crucial step in calculus that makes the results easier to interpret and use. After differentiating, simplifying the resulting expression can reveal fundamental insights about the function's behavior. In our example, simplifying involved reducing \( -\frac{2}{3} x^{-5/3} \) and confirming the expression as neat and readable.

• Check the exponents: combine like terms and carefully consolidate various powers.
• Use fractional properties: manipulate expressions to move between negative and positive exponents if necessary.
• Writing it more presentable: reduce unnecessary negative signs to represent as positive exponents.

An optional but sometimes beneficial step is rewriting expressions with positive exponents for clarity. This insight takes \( -\frac{2}{3} x^{-5/3} \) and represents it as \( -\frac{2}{3} \frac{1}{x^{5/3}} \), offering a clearer perspective on the derivative's behavior.