Problem 27 - Calculus and its applications

Question

Find each derivative. $$ \frac{d}{d x}\left(-2 \sqrt[3]{x^{5}}\right) $$

Step-by-Step Solution

Answer

The derivative is \(-\frac{10}{3} x^{\frac{2}{3}}\).

Step 1: Rewrite the expression

The expression given is \(-2 \sqrt[3]{x^5}\). First, rewrite it in a form that is easier to differentiate by expressing the cube root in terms of exponents: \(-2 (x^5)^{\frac{1}{3}}\). This simplifies to \(-2 x^{\frac{5}{3}}\).

Step 2: Differentiate using the Power Rule

The power rule for differentiation states that the derivative of \(x^n\) is \(n x^{n-1}\). Apply this to the expression \(-2 x^{\frac{5}{3}}\):\(\frac{d}{dx}(-2 x^{\frac{5}{3}}) = -2 \cdot \frac{5}{3} \cdot x^{\frac{5}{3} - 1}\).

Step 3: Simplify the expression

Simplify the derivative obtained from the power rule:\(-2 \cdot \frac{5}{3} \cdot x^{\frac{5}{3} - 1} = -\frac{10}{3} \cdot x^{\frac{2}{3}}\). This expression allows you to see the derivative more clearly.

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus for finding derivatives. It provides a quick and efficient way to differentiate power functions of the form \(x^n\). When using the power rule, you apply the formula \(\frac{d}{dx}(x^n) = n \cdot x^{n-1}\). This rule tells us to multiply the original power \(n\) by the entire expression and then decrease the exponent by one.

• For example, if you have \(x^3\), the derivative would be \(3x^{2}\).
• Similarly, \(x^{-2}\) would become \(-2x^{-3}\) upon differentiation.

In this context, the power rule helps transform complex expressions into more manageable pieces, making differentiation much easier. Knowing how to use the power rule effectively is essential for tackling calculus problems.

Differentiation

Differentiation is the process of finding the derivative of a function. It allows us to evaluate the rate at which a function changes with respect to a variable, typically denoted as \(x\). In simpler terms, it's the means through which we can find the slope of a curve at any point along its path.
Differentiation has various applications such as:

• Finding the velocity in physics.
• Optimizing functions to find maximum or minimum points.
• Determining the rate of change in economics and biology.

The step-by-step process of differentiation usually involves simplifying the function first, as seen with expressions like \(-2 (x^5)^{\frac{1}{3}}\) being rewritten as \(-2 x^{\frac{5}{3}}\) before applying the power rule. Mastery in differentiation opens doors to numerous advanced mathematical applications.

Exponent Rules

Understanding exponent rules is crucial when dealing with functions involving powers since they dictate how we manipulate expressions with exponents. These rules include the basics like:

• \(x^a \cdot x^b = x^{a+b}\)
• \((x^a)^b = x^{a \cdot b}\)
• \(x^{-a} = \frac{1}{x^a}\)

In the provided exercise, turning \(\sqrt[3]{x^5}\) into \(x^{\frac{5}{3}}\) leverages the rule \(x^{a/b}\) means the \(b\)th root of \(x\) to the \(a\)th power. Proper application of these rules simplifies expressions, facilitating the process of differentiation and making the calculations more straightforward. Comprehending exponent rules thoroughly aids in smoothly converting complex radical expressions into their exponential forms, preparing them for the application of the power rule and other differentiation techniques.