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In calculus, the product rule is a handy tool used when you need to find the derivative of the product of two functions. If you have a function that is the product of two simpler functions, let's call them \( f(x) \) and \( g(x) \), the product rule provides a formula for calculating the derivative of their product.

The product rule states that to find the derivative of \( f(x) \times g(x) \), you use the following formula:

• \((f(x)g(x))' = f'(x)g(x) + f(x)g'(x).\)

This means you'll need to calculate the derivatives of both functions, \( f'(x) \) and \( g'(x) \), and then plug these into the formula.
Applying this to the given function \( G(x) = 4x^2(x^3 + 5x) \), recognize the two functions as \( f(x) = 4x^2 \) and \( g(x) = x^3 + 5x \). Calculating their derivatives gives \( f'(x) = 8x \) and \( g'(x) = 3x^2 + 5 \).

Substituting these into the product rule formula, you obtain:

• \[ G'(x) = 8x(x^3 + 5x) + 4x^2(3x^2 + 5) \]

Upon simplifying, you'll find the derivative: \( G'(x) = 20x^4 + 60x^2 \). This approach emphasizes how the product rule decomposes a complex differentiation task into manageable parts.

Simplified differentiation involves simplifying a function first before finding its derivative. For some expressions, this process makes differentiation easier and faster.

In the original problem, the function is expanded before differentiating. The expansion of the product \( 4x^2(x^3 + 5x) \) results in:

• \( G(x) = 4x^5 + 20x^3 \)

By working with the simplified function, you can directly apply the power rule, which is straightforward: if you have \( ax^n \), its derivative is \( anx^{n-1} \).
Differentiating each term separately gives:

• The derivative of \( 4x^5 \) is \( 20x^4 \)
• The derivative of \( 20x^3 \) is \( 60x^2 \)

Adding these results gives \( G'(x) = 20x^4 + 60x^2 \).

This simplified method provides a quick way to reach the answer, especially when dealing with polynomial expressions. It reinforces understanding of basic differentiation rules and their application.

A graphing calculator can be a valuable tool for verifying the result of a derivative, especially in complex expressions.
For this exercise, once you've derived that \( G'(x) = 20x^4 + 60x^2 \), use a graphing calculator to check the correctness of your work. Here’s how:

• First, enter the original function \( G(x) = 4x^2(x^3 + 5x) \) into the calculator.
• Use the calculator's differentiation feature to compute the derivative, or manually calculate the derivative at various points.
• Verify that the derivative matches \( 20x^4 + 60x^2 \) by comparing the calculator's results with your calculations.

You might also graph both the original function and the calculated derivative. Observe the slope of the original function graph at different points; it should align with the slope given by the derivative graph.
Graphing calculators not only validate your manual calculations but also help visualize the concepts of calculus, reinforcing understanding of how derivatives work in practical scenarios. This approach provides a digital confirmation, marrying manual math with tech-assisted learning.