Problem 17 - Brief Applied Calculus

Question

Find the derivative of each function. $$ f(x)=\frac{6}{\sqrt[3]{x}} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -\frac{2}{\sqrt[3]{x^4}} \).

Step 1: Rewrite the function

The function given is \( f(x)=\frac{6}{\sqrt[3]{x}} \). To make it easier to differentiate, rewrite the cube root in the denominator as an exponent. Recall that \(\sqrt[3]{x} = x^{1/3}\). Thus, the function becomes \( f(x) = 6x^{-1/3} \).

Step 2: Differentiate using the power rule

To find the derivative of \( f(x) = 6x^{-1/3} \), apply the power rule, which states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). Here, \( n = -1/3 \). Therefore, the derivative \( f'(x) \) is computed as follows:\[ f'(x) = 6 \cdot \left(-\frac{1}{3}\right) x^{-1/3 - 1} \]Simplifying, we get: \[ f'(x) = -2 x^{-4/3} \]

Step 3: Express the derivative in radical form

Rewrite the derivative \( f'(x) = -2 x^{-4/3} \) using radical notation to simplify the expression:\[ f'(x) = -\frac{2}{x^{4/3}} = -\frac{2}{\sqrt[3]{x^4}} \]Thus, the derivative in radical form is \( f'(x) = -\frac{2}{\sqrt[3]{x^4}} \).

Key Concepts

Derivative

In calculus, the derivative represents the rate of change of a function relative to changes in its variable. This is useful for understanding how the function behaves as the input changes. Think of it as the slope of the tangent line to the curve of the function at any given point. A derivative is denoted as \( f'(x) \) or \( \frac{df}{dx} \), indicating how the function \( f(x) \) changes with respect to \( x \).

The process of finding a derivative is called differentiation. There are specific rules and techniques that aid in such calculations, like the power rule or product rule. For instance, in our exercise, we are tasked to find the derivative of the function \( f(x) = \frac{6}{\sqrt[3]{x}} \). First, converting the function to exponent form makes it simpler to differentiate. Then, by applying those techniques, we determine how the function changes across different values of \( x \).

Power Rule

The power rule is an essential tool in calculus for finding derivatives quickly and easily. It states that if you have a function \( f(x) = x^n \), its derivative is \( f'(x) = n x^{n-1} \). This is extremely helpful when working with polynomials or any function where the variable is raised to a power.

In our example, after rewriting the original function, it becomes \( f(x) = 6x^{-1/3} \). Applying the power rule here involves multiplying the exponent by the coefficient (the \( 6 \) in this case) and reducing the power by one, hence \( f'(x) = -2 x^{-4/3} \).

• The coefficient, \( 6 \), is multiplied by the new exponent, \( -1/3 \).
• The exponent is then decreased by one to form the new power, \( -4/3 \).

Using this rule greatly simplifies the process of differentiation, avoiding complex algebraic manipulations.

Radical Notation

Radical notation involves expressing numbers and functions in terms of roots, such as square roots or cube roots. It’s a convenient way to write these expressions compactly. For instance, the cube root of \( x \) is written as \( \sqrt[3]{x} \).

In the given exercise, after finding the derivative in terms of exponents, it’s often advantageous to revert it back to radical form for simplification or interpretation. The derivative \( -2 x^{-4/3} \), when re-expressed in radical notation, becomes \( -\frac{2}{\sqrt[3]{x^4}} \).

• Radical notation helps in visualization and understanding the behavior of expressions, especially those involving powers.
• Switching between exponent and radical form is a common practice to choose the most useful representation depending on the particular problem or context.

Understanding both forms enhances flexibility in problem-solving scenarios.