Problem 30 - Calculus and its applications

Question

Differentiate each function $$ G(x)=\sqrt[3]{x^{5}+6 x} $$

Step-by-Step Solution

Answer

The derivative is \( G'(x) = \frac{1}{3}(x^5 + 6x)^{-2/3} \cdot (5x^4 + 6) \).

Step 1: Rewrite the Function

The given function is \( G(x) = \sqrt[3]{x^5 + 6x} \). To differentiate it, rewrite the function using exponent notation: \( G(x) = (x^5 + 6x)^{1/3} \). This makes it easier to apply differentiation rules.

Step 2: Differentiate Using the Chain Rule

Apply the chain rule to differentiate the function \( G(x) = (x^5 + 6x)^{1/3} \). The chain rule states that if you have a composite function \( g(f(x)) \), then the derivative \( g'(f(x)) \) is \( g'(f(x)) \cdot f'(x) \). Here, the outer function is \( u^{1/3} \) and the inner function is \( u = x^5 + 6x \).

Step 3: Find the Derivative of the Outer Function

Differentiate \( u^{1/3} \) with respect to \( u \). The derivative of \( u^{1/3} \) is \( \frac{1}{3}u^{-2/3} \) according to the power rule.

Step 4: Find the Derivative of the Inner Function

Differentiate the inside function \( x^5 + 6x \) with respect to \( x \). Using the power rule, \( d/dx(x^5) = 5x^4 \), and for the linear term, \( d/dx(6x) = 6 \). So the derivative of the inner function is \( 5x^4 + 6 \).

Step 5: Apply the Chain Rule

Combine the derivatives using the chain rule: multiply the derivative of the outer function \( \frac{1}{3}(x^5 + 6x)^{-2/3} \) by the derivative of the inner function \( 5x^4 + 6 \).

Step 6: Simplify the Expression

The derivative \( G'(x) \) is given by: \[ G'(x) = \frac{1}{3}(x^5 + 6x)^{-2/3} \cdot (5x^4 + 6) \]. You may leave it in this form, or factor further if necessary.

Key Concepts

Chain Rule

The chain rule is an essential technique when dealing with composite functions. Composite functions are functions where one function is nested inside another. In this exercise, we have the function \( G(x) = (x^5 + 6x)^{1/3} \). Here, the outer function is \( u^{1/3} \), where \( u = x^5 + 6x \).
To use the chain rule effectively, follow these steps:

• Identify the outer and inner functions in your composite function.
• Differentiate the outer function with respect to the inner function.
• Next, differentiate the inner function with respect to \( x \).
• Finally, multiply these derivatives together to find the derivative of the composite function.

The chain rule is expressed as \( (g(f(x)))' = g'(f(x)) \cdot f'(x) \). This approach allows us to peel layers of nested functions effectively, ensuring we capture all changes occurring within the function.

Power Rule

The power rule is fundamental in calculus, especially when differentiating polynomial functions. This rule is straightforward: if you have a function \( f(x) = x^n \), the derivative is \( f'(x) = n \cdot x^{n-1} \). This rule allows us to quickly find derivatives of powers of \( x \).
In our exercise, we apply this rule multiple times. First, to find the derivative of the inner function \( x^5 + 6x \), we apply the power rule to each term:

• For \( x^5 \), the derivative becomes \( 5x^4 \).
• For \( 6x \), the derivative is the constant \( 6 \).

For the outer function \( u^{1/3} \), after substituting \( u = x^5 + 6x \), the power rule again helps us find that the derivative is \( \frac{1}{3}u^{-2/3} \). This demonstrates the flexibility and usefulness of the power rule when combined with other differentiation techniques such as the chain rule.

Function Derivative

Understanding how to compute a function's derivative is crucial for analyzing how functions behave, as well as solving real-world problems involving rates of change. In this exercise, we differentiated the function \( G(x) = \sqrt[3]{x^5 + 6x} \).
Here's a simple breakdown of how we approached this:

• First, rewrite the function using exponent notation to facilitate differentiation. Here, that meant changing \( \sqrt[3]{x^5 + 6x} \) to \( (x^5 + 6x)^{1/3} \).
• Apply the chain rule to handle the composition of the functions, identifying and differentiating the outer and inner functions separately.
• Use the power rule to differentiate each part, making it easier to calculate the overall derivative efficiently.
• Finally, combine the derivatives to find the complete derivative, resulting in \[ G'(x) = \frac{1}{3}(x^5 + 6x)^{-2/3} \cdot (5x^4 + 6) \].

Derivatives tell us how a function's output changes as the input changes, providing insights into the function's characteristics and behavior over its domain.