Textbook Question
7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
y = x² - a² / x-a, where a is a constant
3. Techniques of Differentiation
Product and Quotient Rules
3:20 minutesProblem 3.85c
Textbook Question
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 guidance1
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).
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.
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).
Step 4: Calculate each term separately. First, compute f'(3)g(3) and then f(3)g'(3).
Step 5: Add the results from Step 4 to find the derivative of the product at x = 3.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
05:18
The Product Rule
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.
Recommended video:
04:50Critical Points
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.
Recommended video:
05:44Derivatives
Related Videos
Related Practice