Problem 10 - The Calculus with Analytic Geometry

Question

Find the derivative of the given function. $$ f(x)=3 x^{2 / 3}-6 x^{1 / 3}+x^{-1 / 3} $$

Step-by-Step Solution

Answer

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

Step 1 - Identify the function to differentiate

The function given is \( f(x) = 3x^{2/3} - 6x^{1/3} + x^{-1/3} \).

Step 2 - Use the power rule for differentiation

Apply the power rule to each term of the function. The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\).

Step 3 - Differentiate the first term

Differentiate \(3x^{2/3}\): \[ \frac{d}{dx}(3x^{2/3}) = 3 \cdot \frac{2}{3}x^{2/3 - 1} = 2x^{-1/3} \]

Step 4 - Differentiate the second term

Differentiate \(-6x^{1/3}\): \[ \frac{d}{dx}(-6x^{1/3}) = -6 \cdot \frac{1}{3}x^{1/3 - 1} = -2x^{-2/3} \]

Step 5 - Differentiate the third term

Differentiate \(x^{-1/3}\): \[ \frac{d}{dx}(x^{-1/3}) = -\frac{1}{3}x^{-1/3 - 1} = -\frac{1}{3}x^{-4/3} \]

Step 6 - Combine the results from all terms

Summing the derivatives of each term gives: \[ f'(x) = 2x^{-1/3} - 2x^{-2/3} - \frac{1}{3}x^{-4/3} \]

Key Concepts

The Power Rule

To understand how to find the derivative of a function like the one in the given exercise, it's essential to grasp the power rule. The power rule is a fundamental principle in calculus used for differentiation. It simplifies the process of finding derivatives of power functions. If you have a function in the form of \(x^n\), where \(n\) is any real number, its derivative is found by multiplying the exponent \(n\) by the base \(x\) raised to the power of \(n-1\). In mathematical terms, the power rule states that: \[ \frac{d}{dx}x^n = nx^{n-1} \]
This rule applies regardless of whether \(n\) is positive, negative, or a fraction. For example:

• For \(x^{3}\), the derivative is \(3x^{2}\).
• For \(x^{-1}\), the derivative is \(-x^{-2}\).
• For \(x^{1/2}\), the derivative is \(\frac{1}{2}x^{-1/2}\).

By using the power rule, we can quickly find derivatives without dealing with limits or more complex differentiation techniques.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents the rate at which a function is changing at any point. It gives us the slope of the function's graph at that point.
In the context of the exercise, differentiation involves applying the power rule to each term of the function \(f(x) = 3x^{2/3} - 6x^{1/3} + x^{-1/3}\). Here’s a step-by-step breakdown:

• First term: Differentiate \(3x^{2/3}\). Using the power rule: \[ \frac{d}{dx}(3x^{2/3}) = 3 \cdot \frac{2}{3}x^{2/3 - 1} = 2x^{-1/3} \]
• Second term: Differentiate \(-6x^{1/3}\). Using the power rule: \[ \frac{d}{dx}(-6x^{1/3}) = -6 \cdot \frac{1}{3}x^{1/3 - 1} = -2x^{-2/3} \]
• Third term: Differentiate \(x^{-1/3}\). Using the power rule: \[ \frac{d}{dx}(x^{-1/3}) = -\frac{1}{3}x^{-1/3 - 1} = -\frac{1}{3}x^{-4/3} \]

Combining these results gives the derivative: \( f'(x) = 2x^{-1/3} - 2x^{-2/3} - \frac{1}{3}x^{-4/3} \).

Derivative of Power Function

A power function is any function of the form \(f(x) = kx^n\), where \(k\) and \(n\) are constants. Calculating the derivative of a power function always involves applying the power rule.
In the given exercise, we had to find the derivatives of three power functions within the larger function. Here's a quick recap of how it’s done: 1. Identify the term to be differentiated.
2. Apply the power rule: Multiply the power by the coefficient and subtract one from the exponent.
Let’s revisit our functions:

• For \(3x^{2/3}\), we multiplied the coefficient 3 by the exponent \(2/3\) giving us \(2\), and then subtracted 1 from \(2/3\) resulting in \(-1/3\). Hence, the derivative is \(2x^{-1/3}\).
• For \(-6x^{1/3}\), we multiplied the coefficient -6 by the exponent \(1/3\) giving us \(-2\), and then subtracted 1 from \(1/3\) resulting in \(-2/3\). Hence, the derivative is \(-2x^{-2/3}\).
• For \(x^{-1/3}\), we multiplied the coefficient 1 by the exponent \(-1/3\) giving us \(-1/3\), and then subtracted 1 from \(-1/3\) resulting in \(-4/3\). Hence, the derivative is \(-1/3)x^{-4/3}\).

The power rule makes it straightforward to find these derivatives, provided you carefully follow each step.