Problem 25 - Brief Applied Calculus

Question

Find the derivative of each function. $$ h(x)=6 \sqrt[3]{x^{2}}-\frac{12}{\sqrt[3]{x}} $$

Step-by-Step Solution

Answer

The derivative is \( h'(x) = 4x^{1/3} + 4x^{-4/3} \).

Step 1: Rewrite in Exponential Form

To differentiate the function, first, rewrite the radicals in exponential form. The function is given as \[ h(x) = 6 \sqrt[3]{x^2} - \frac{12}{\sqrt[3]{x}}. \] Rewrite it to \[ h(x) = 6 (x^2)^{1/3} - 12 x^{-1/3}. \] This will make it easier to utilize the power rule for differentiation.

Step 2: Apply the Power Rule

Utilize the power rule \( \frac{d}{dx} [x^n] = n \cdot x^{n-1} \) to differentiate each term:1. For the first term, \( 6 (x^2)^{1/3} \), use the chain rule: - Differentiate: \( 6 \cdot \frac{2}{3} x^{\frac{2}{3} - 1} \cdot d(x^2)/dx \) - Which equals: \( 6 \cdot \frac{2}{3} x^{-\frac{1}{3}} \cdot 2x = 4x^{1/3}. \)2. For the second term, \( -12x^{-1/3} \), differentiate directly: - Differentiate: \( -12 \cdot (-\frac{1}{3}) x^{-\frac{1}{3} - 1} \) - Which simplifies to: \( 4x^{-4/3}. \)

Step 3: Simplify the Derivative

Combine the results from differentiating each term. First term yielded \( 4x^{1/3} \) and the second term yielded \( 4x^{-4/3} \). Therefore, the derivative is:\[ h'(x) = 4x^{1/3} + 4x^{-4/3}. \] This is the simplified expression for the derivative of the given function.

Key Concepts

Power Rule

When finding the derivative of a function, the power rule is one of the easiest and most essential tools to use. It states that for any term in the form of \( x^n \), the derivative is \( n \cdot x^{n-1} \).
This is straightforward because it involves multiplying the original exponent by the coefficient and reducing the exponent by one. In the given exercise, you used this rule to tackle the terms in their exponential form, allowing you to differentiate efficiently. For instance, consider the term \( -12x^{-1/3} \).
Applying the power rule here lets you obtain \(-12 \cdot \left(-\frac{1}{3}\right) x^{-4/3} \). This simplification greatly speeds up the process.
The power rule is powerful since it adheres to

• Efficiency: Quickly compute derivatives of polynomial terms.
• Versatility: Apply to positive or negative exponents, including fractions.

Chain Rule

The chain rule is another crucial differentiation technique, especially when dealing with composite functions. A composite function is when a function is nested within another function, such as \((x^2)^{1/3}\) in our exercise.
The chain rule indicates that you should differentiate the outer function, multiply it by the derivative of the inner function. For \(6 (x^2)^{1/3}\), you start by differentiating the outer function, which involves the power rule: \(\frac{1}{3}(x^2)^{-2/3}\). Then multiply it by the derivative of the inner function \(x^2\), leading to \(2x\).
This gives:

• Outer derivative: \(\frac{1}{3}(x^2)^{-2/3}\) turns to \(x^{-2/3}\).
• Inner derivative: \(2x\), resulting in \(6 \cdot \frac{2}{3} x^{-1/3} \cdot 2x\).

The chain rule helps massively by ensuring no step in differentiating nested functions is skipped.

Exponential Form

Exponential form is particularly handy for converting radical expressions to something easier to differentiate. When faced with expressions containing roots, such as the cube roots in the exercise, rewriting these in exponential terms simplifies the process. For example,
the cube root of \(x^2\) is efficiently written as \((x^2)^{1/3}\). By converting everything:

• \(6 \sqrt[3]{x^2}\) becomes \(6(x^2)^{1/3}\)
• \(-\frac{12}{\sqrt[3]{x}}\) becomes \(-12x^{-1/3}\)

This reformulation makes it straightforward to apply standard differentiation rules like the power rule.
Thus, always consider switching complicated root expressions into exponential form for solving problems more easily.