Question

Compute the derivative of the given function. $$f(x)=\sqrt[3]{x}+x^{2 / 3}$$

Short answer

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

Step-by-step solution

Express the Function in Power Form

Before differentiating, rewrite the function in a form that's easier to differentiate. The given function is \( f(x) = \sqrt[3]{x} + x^{2/3} \). Express \( \sqrt[3]{x} \) as \( x^{1/3} \). Now, the function becomes \( f(x) = x^{1/3} + x^{2/3} \).

Apply the Power Rule

To differentiate the function \( f(x) = x^{1/3} + x^{2/3} \), use the power rule, which states that \( \frac{d}{dx}x^n = nx^{n-1} \).

Differentiate Each Term

Differentiate each term of \( f(x) \):- The derivative of \( x^{1/3} \) is \( \frac{1}{3}x^{-2/3} \).- The derivative of \( x^{2/3} \) is \( \frac{2}{3}x^{-1/3} \).

Combine the Derivatives

Combine the derivatives from Step 3:\( f'(x) = \frac{1}{3}x^{-2/3} + \frac{2}{3}x^{-1/3} \).

Rewrite the Derivative

Express the derivative in terms of fractional exponents:- \( f'(x) = \frac{1}{3}\frac{1}{x^{2/3}} + \frac{2}{3}\frac{1}{x^{1/3}} \) (optional).This simplifies to \( f'(x) = \frac{1}{3x^{2/3}} + \frac{2}{3x^{1/3}} \).

Key concepts

Power Rule

The power rule is a fundamental tool in calculus used for differentiation. It states that for any function in the form of \( x^n \), its derivative is \( nx^{n-1} \). This simplifies the process of finding the slope of a tangent line to a curve at any given point.
In the context of our exercise, we use the power rule to differentiate each term of the given function. By applying it consistently, we transform each term from its original form to its derivative.

• The power rule helps simplify complex functions into manageable parts.
• It provides a quick way to find the rate of change of functions expressed in polynomial form.
• It is essential for solving a variety of calculus problems involving curves and slopes.

Understanding and practicing the power rule will make differentiation tasks simpler and less intimidating.

Fractional Exponents

Fractional exponents represent roots of numbers in a more algebraic expression. For instance, the cube root of x, written as \( \sqrt[3]{x} \), can be represented as \( x^{1/3} \). This conversion is crucial for applying the power rule.
When dealing with fractional exponents:

• Converting roots into exponents simplifies differentiation.
• Each fractional exponent \( x^{m/n} \) means the nth root of x raised to the mth power.
• Fractional exponents can often make calculations clearer and more concise.

Grasping this concept allows you to link algebra with calculus, making transition to calculus concepts smoother.

Differentiation

Differentiation is the process of finding the derivative of a function, which tells us the rate at which a function is changing at any point. It's a core concept in calculus that extends into various applications, including physics and economics.
The steps for differentiation typically include:

• Rewriting the function in a differentiable form, like converting roots to fractional exponents.
• Applying the power rule to find the derivatives of each term in the function.
• Combining the derivatives to express the overall rate of change.

This concept is at the heart of calculus, allowing us to predict and understand behaviors of different mathematical models.

Mathematics Education

Learning mathematics is more than just memorizing formulas or solving equations. It's about understanding patterns, developing problem-solving skills, and applying these skills in real-world scenarios.
In the realm of calculus:

• Concepts like the power rule and differentiation form the backbone of many scientific and engineering problems.
• Fractional exponents help bridge the gap between algebra and calculus.
• Education in mathematics should emphasize comprehension over rote learning to cultivate these abilities.

To foster a deeper understanding, mathematics education should encourage exploration and application of learned concepts, laying a foundation for lifelong learning and adaptation.