Problem 9 - Brief Applied Calculus

Question

Find the derivative of each function. $$ g(w)=6 \sqrt[3]{w} $$

Step-by-Step Solution

Answer

The derivative of \( g(w) = 6 \sqrt[3]{w} \) is \( \frac{2}{w^{2/3}} \).

Step 1: Express the Function in Exponential Form

The given function is \( g(w) = 6 \sqrt[3]{w} \). We can rewrite it using exponents as \( g(w) = 6w^{1/3} \) because the cube root of \( w \) is equivalent to raising it to the power of \( 1/3 \).

Step 2: Differentiate Using the Power Rule

To find the derivative of \( g(w) = 6w^{1/3} \), we apply the power rule for derivatives, which states that \( \frac{d}{dw} [w^n] = nw^{n-1} \). Here, \( n = \frac{1}{3} \).

Step 3: Apply the Constant Multiple Rule

Since the function has a constant factor of 6, apply the constant multiple rule: \( \frac{d}{dw}[cf(w)] = c \cdot \frac{d}{dw}[f(w)] \). Thus, \( \frac{d}{dw}[6w^{1/3}] = 6 \cdot \frac{1}{3}w^{1/3-1} \).

Step 4: Simplify the Derivative Expression

Simplify the expression from the previous step: \( 6 \cdot \frac{1}{3} w^{-2/3} = 2w^{-2/3} \).

Step 5: Rewrite the Derivative in a Simplified Form

Finally, we can express the derivative using positive exponents: \( 2w^{-2/3} = \frac{2}{w^{2/3}} \). Therefore, the derivative of \( g(w) \) is \( \frac{2}{w^{2/3}} \).

Key Concepts

Power Rule

The power rule is a fundamental technique in calculus for finding derivatives. It's a handy rule that streamlines the process when you're working with functions involving powers of a variable. Simply put, if you have a function of the form \( f(x) = x^n \), the derivative is given by \( f'(x) = nx^{n-1} \). This means you bring down the exponent as a coefficient in front of the variable and then decrease the original exponent by one.

For example, consider the function \( g(w) = 6w^{1/3} \). Using the power rule, focus on the segment \( w^{1/3} \). The power here is \( \frac{1}{3} \), so the derivative \( \frac{d}{dw} [w^{1/3}] \) becomes \( \frac{1}{3}w^{-2/3} \). By integrating the power rule with other rules such as the constant multiple rule, you can find derivatives of more complex functions quickly.

Exponential Form

When dealing with derivatives, it's often beneficial to rewrite functions in exponential form. This approach makes differentiation more straightforward, especially when dealing with roots or similar operations.

Take \( g(w) = 6 \sqrt[3]{w} \). The cube root can be rewritten in exponential form as \( w^{1/3} \). This is because roots can be represented as fractional exponents, where the root index becomes the denominator. In general, \( \sqrt[n]{x} = x^{1/n} \). For differentiation, expressing roots as exponents allows you to harness the power rule seamlessly.

Switching to exponential form converts functions to a more "calculus-friendly" version, enabling the use of concise rules like the power rule to compute derivatives. It's all about making functions easier to handle analytically.

Constant Multiple Rule

The constant multiple rule is a simple yet powerful rule for differentiation. When a function is multiplied by a constant, the rule states that the derivative of the entire function is that constant multiplied by the derivative of the function itself.

Mathematically, if \( f(w) = c \, g(w) \), then \( f'(w) = c \, g'(w) \). For instance, in the function \( g(w) = 6w^{1/3} \), the constant \( 6 \) multiplies the function \( w^{1/3} \). When differentiating, keep the constant \( 6 \) and apply it to the derivative of \( w^{1/3} \), thus \( \frac{d}{dw}[6w^{1/3}] = 6 \cdot \frac{1}{3}w^{-2/3} \).

This rule enables simplification and speed when a constant factor is involved, preventing errors in calculations.

Simplifying Derivatives

Once you've computed a derivative, simplifying it is often necessary. Simplification aids clarity, especially when presenting your result in a more understandable or standardized form.

For example, after applying calculus rules to \( 6w^{1/3} \), we derive \( 6 \cdot \frac{1}{3} w^{-2/3} \). Calculating this gives \( 2w^{-2/3} \). To further simplify: instead of negative exponents, rewrite them as fractions, resulting in \( \frac{2}{w^{2/3}} \).

This final form is clearer and better suited for many applications, especially those requiring positive indices. Simplifying derivatives is a crucial step beyond computation, ensuring results are both correct and easily interpretable.