Problem 6 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Function Form

The function given is \( f(x) = x^{1/3} \), which is a power function. To differentiate it, we will apply the power rule for derivatives.

Step 2: Apply the Power Rule

The power rule states that if \( f(x) = x^n \) then \( f'(x) = nx^{n-1} \). Here, \( n = \frac{1}{3} \). Using the power rule, we have: \( f'(x) = \frac{1}{3}x^{\frac{1}{3}-1} \).

Step 3: Simplify the Exponent

Subtract \( 1 \) from \( \frac{1}{3} \). This gives us \( \frac{1}{3} - 1 = \frac{1}{3} - \frac{3}{3} = -\frac{2}{3} \).

Step 4: Write the Simplified Derivative

The simplified form of the derivative is \( f'(x) = \frac{1}{3}x^{-\frac{2}{3}} \).

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus that helps in finding the derivative of power functions easily. Understanding this rule is essential for anyone looking to master differentiation. Here’s a simple way to remember the power rule:

• If you have a function of the form \( f(x) = x^n \), where \( n \) is any real number, the power rule states that the derivative \( f'(x) \) is \( nx^{n-1} \).
• In simpler terms, you multiply the original power by the coefficient and then reduce the power by one.

For example, if you are differentiating \( x^5 \), by applying the power rule, you get \( 5x^{5-1} = 5x^4 \). This rule is extremely helpful for quickly finding derivatives and is widely used in various mathematical applications.
Let's introduce an important point: the power rule applies to all real number exponents, including fractions and negative numbers.

Derivative of Power Functions

A power function is any function of the form \( f(x) = x^n \) where \( n \) is a constant. Calculating its derivative requires applying the power rule. Let's look at an example with the function \( f(x) = x^{1/3} \):

• First, identify \( n \), which in this case is \( \frac{1}{3} \).
• Then, apply the power rule: \( f'(x) = \frac{1}{3}x^{\frac{1}{3}-1} \).

Now, it's just a matter of simplifying the expression to understand it better. It's crucial to remember that finding the derivative tells you the rate at which the function's value changes with respect to \( x \), making it a powerful tool in calculus.
Recognizing and differentiating power functions is a skill that will keep appearing throughout your calculus journey. It emphasizes understanding how exponents and operations on them work in calculus.

Simplification of Exponents

Simplifying exponents is an essential step in obtaining the derivative in its cleanest form. This involves adjusting the exponent after applying the power rule:

• When you have an exponent such as \( \frac{1}{3} - 1 \), you perform the subtraction like you would with fractions.
• Convert \( 1 \) into a fraction: \( \frac{3}{3} \). Subtracting gives \( \frac{1}{3} - \frac{3}{3} = -\frac{2}{3} \).

At this stage, the derivative is expressed in a simplified manner: \( f'(x) = \frac{1}{3}x^{-\frac{2}{3}} \).
Writing exponents in their simplest form not only makes the derivative look cleaner but also helps in further calculations and interpretations, especially in complex problems.
By mastering the art of simplifying exponents, you solidify your understanding of the entire differentiation process.