Understanding the Product Rule for Derivatives

This video tutorial covers the product rule for derivatives, providing multiple examples to illustrate its application. It begins with an introduction to the rule, followed by examples involving polynomial and trigonometric functions. The tutorial also demonstrates how to handle expressions with multiple factors and evaluates derivatives at specific points using given values.

What is the product rule formula for the derivative of two functions f and g?

• f * g + f' * g'

• f * g' + f' * g'

• f' * g + f * g'

• f' * g' + f * g

In the example of differentiating x^2 * sin(x), what is the derivative of x^2?

• 2x

• x

• x^2

• 2x^2

When applying the product rule to 5x - 9x^3 and 8 + x^2, what is the derivative of 9x^3?

• 27x^2

• 9x^2

• 3x^2

• 18x^2

In the expression 4sin(x)tan(x), what is the derivative of tan(x)?

• sec(x)

• cos(x)

• sec^2(x)

• sin(x)

How do you apply the product rule to the expression 5xsin(x) - x^3tan(x)?

• Differentiate both terms together

• Use the product rule on each term separately

• Ignore the trigonometric functions

• Only differentiate the polynomial terms

What is the derivative of x^3 in the expression x^3tan(x)?

• 3x^2

• x^2

• 3x

• x^3

In the example with three factors, what is the first step in applying the product rule?

• Differentiate all factors simultaneously

• Differentiate the middle factor

• Differentiate the first factor

• Differentiate the last factor

What is the derivative of the constant term in the expression x^2 + 6?

• x

• 1

• 0

• 6

How do you evaluate f'g + fg' at x = 3 given f(3) = 4, f'(3) = -7, g(3) = -8, and g'(3) = 5?

• Subtract f'(3) from g(3) and add f(3) times g'(3)

• Add f(3) and g'(3) and multiply by f'(3) and g(3)

• Multiply f(3) by g'(3) and add f'(3) times g(3)

• Multiply f'(3) by g'(3) and add f(3) times g(3)

What is the result of evaluating f'g + fg' at x = 3 with the given values?

• 96

• 36

• 76

• 56