Understanding the Product Rule in Differentiation

The video tutorial reviews the product rule of differentiation, explaining that the derivative of a product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first. The tutorial applies this rule to find the derivative of a specific function, f(x) = 3x^4 * cos(x), by identifying the components U and V, calculating their derivatives, and then using the product rule. Finally, the result is factored to simplify the expression.

What does the product rule of differentiation state?

• The derivative of a quotient is the quotient of the derivatives.

• The derivative of a product is the first function times the derivative of the second plus the second function times the derivative of the first.

• The derivative of a product is the product of the derivatives.

• The derivative of a sum is the sum of the derivatives.

In the example f(x) = 3x^4 * cos(x), what is the derivative of U, where U = 3x^4?

• 12x^3

• 3x^3

• 4x^3

• 12x^4

What is the derivative of V, where V = cos(x), in the example?

• -sin(x)

• cos(x)

• sin(x)

• -cos(x)

After applying the product rule to f(x) = 3x^4 * cos(x), what is the expression for f'(x) before factoring?

• 3x^4 * sin(x) - cos(x) * 12x^3

• 3x^4 * cos(x) + sin(x) * 12x^3

• 3x^4 * cos(x) - sin(x) * 12x^3

• 3x^4 * sin(x) + cos(x) * 12x^3

What is the factored form of the derivative f'(x) for the function f(x) = 3x^4 * cos(x)?

• 3x^3(sin(x) + x cos(x))

• 3x^3(cos(x) - x sin(x))

• 3x^3(sin(x) - x cos(x))

• 3x^3(cos(x) + x sin(x))