91Ó°ÊÓ

The product rule is a handy tool when you're dealing with the derivative of a product involving two differentiable functions. To remember this, think of the formula as the combination where each function takes a turn, together with its derivative counterpart. Introducing the rule: For two functions, \( f(x) \) and \( g(x) \), the derivative of their product \( h(x) = f(x)g(x) \), is given by:

• \( h'(x) = f'(x)g(x) + f(x)g'(x) \)

In simpler terms, you multiply the derivative of the first function by the second function, then add it to the product of the first function and the derivative of the second one.

In the exercise, we use the given values from the table at \( x = 2 \) for \( f(2) = 3 \), \( g(2) = 4 \), \( f'(2) = 5 \), and \( g'(2) = -2 \). Plug these into the product rule formula:

• \( h'(2) = (5)(4) + (3)(-2) \)
• \( h'(2) = 20 - 6 = 14 \)

This tells us that the derivative of the product of the two functions at \( x = 2 \) is 14.

The sum rule is one of the simplest derivative rules to understand and apply. This rule applies when you have the sum of two functions and you want to find the derivative of the entire expression. For functions \( f(x) \) and \( g(x) \), the rule states:

• \( h(x) = f(x) + g(x) \)
• The derivative, \( h'(x) = f'(x) + g'(x) \)

It's straightforward: simply add the derivative of each function together to get the derivative of their sum.

Using the table values at \( x = 2 \), where \( f'(2) = 5 \) and \( g'(2) = -2 \), we substitute directly into the sum rule:

• \( h'(2) = 5 + (-2) \)
• \( h'(2) = 3 \)

Thus, the derivative of the sum of the two functions at \( x = 2 \) is 3.

The quotient rule comes into play when differentiating the division of two functions. The formula may look a bit more complex, but breaking it into parts can help. Here's how it works for functions \( f(x) \) and \( g(x) \) where \( h(x) = \frac{f(x)}{g(x)} \):

• The derivative, \( h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \)

Think of it as: "low \( g(x) \)\, d high \( f'(x) \) \- high \( f(x) \)\, d low \( g'(x) \) all over the square of what's below."

From the table for \( x = 2 \), using the values \( f(2) = 3 \), \( g(2) = 4 \), \( f'(2) = 5 \), and \( g'(2) = -2 \), we substitute into our rule:

• \( h'(2) = \frac{(5)(4) - (3)(-2)}{4^2} \)
• \( h'(2) = \frac{20 + 6}{16} = \frac{26}{16} = \frac{13}{8} \)

Therefore, the derivative of the quotient function at \( x = 2 \) is \( \frac{13}{8} \).