Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>


c. d/dx ((f(x)g(x)) |x=3

Verified step by step guidance

1. Step 1: Recall the product rule for derivatives, which states that if you have two functions f(x) and g(x), the derivative of their product is given by (f(x)g(x))' = f'(x)g(x) + f(x)g'(x).
2. Step 2: Identify the values you need from the table. You will need f(3), g(3), f'(3), and g'(3) to apply the product rule at x = 3.
3. Step 3: Substitute the values from the table into the product rule formula. This means replacing f(x) with f(3), g(x) with g(3), f'(x) with f'(3), and g'(x) with g'(3).
4. Step 4: Calculate each term separately. First, compute f'(3)g(3) and then f(3)g'(3).
5. Step 5: Add the results from Step 4 to find the derivative of the product at x = 3.

Key Concepts

Product Rule

The Product Rule is a fundamental differentiation rule used to find the derivative of the product of two functions. It states that if you have two functions, f(x) and g(x), the derivative of their product is given by d/dx(f(x)g(x)) = f'(x)g(x) + f(x)g'(x). This rule is essential for solving problems involving the multiplication of functions.

Evaluating Derivatives at a Point

Evaluating derivatives at a specific point involves substituting the value of x into the derivative expression after it has been calculated. In this case, after applying the Product Rule, you will substitute x = 3 into the resulting expression to find the specific value of the derivative at that point. This step is crucial for obtaining numerical answers from derivative expressions.

Using Derivative Tables

Derivative tables provide pre-calculated values of derivatives for various functions at specific points. When solving derivative problems, such as the one presented, these tables can be used to quickly find f'(3) and g'(3) without needing to compute the derivatives from scratch. This efficiency is particularly useful in problems where time or complexity is a factor.