Understanding the Product Rule of Differentiation

This video tutorial provides a detailed proof of the product rule of differentiation. It begins with an introduction to the rule, stating that the derivative of a product of two functions is the sum of the first function times the derivative of the second and the second function times the derivative of the first. The proof is constructed using the limit definition of the derivative, transforming the expression into a form that allows for factoring and simplification. The tutorial concludes by evaluating the limits to arrive at the final proof, reinforcing the understanding of the product rule.

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 product of the derivatives.

• The derivative of a sum is the sum 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.

What is the first step in applying the limit definition to the product of two functions?

• Add a constant to the functions.

• Apply the limit as h approaches zero.

• Subtract the functions from each other.

• Multiply the functions directly.

Why do we add and subtract the same term when rewriting the expression?

• To form an equivalent fraction.

• To eliminate the denominator.

• To simplify the expression.

• To change the value of the expression.

What is the purpose of factoring common terms in the expression?

• To change the order of operations.

• To eliminate the need for limits.

• To simplify the expression and prepare it for limit evaluation.

• To increase the complexity of the expression.

What is the result of evaluating the limit as h approaches zero for f(x + h)?

• g(x)

• f'(x)

• 0

• f(x)

What is the difference quotient for g(x) as h approaches zero?

• g'(x)

• g(x)

• f'(x)

• f(x)

What does the limit of a product become?

• A difference of limits

• A quotient of limits

• A product of limits

• A sum of limits

What is the final expression for the derivative of f(x)g(x)?

• f(x)g(x)

• f'(x)g'(x)

• f(x)g'(x) + g(x)f'(x)

• f'(x)g(x) - g'(x)f(x)

What is the significance of the product rule in calculus?

• It simplifies the process of finding derivatives of sums.

• It provides a method to find derivatives of products of functions.

• It is used to find integrals of functions.

• It is only applicable to constant functions.

What happens to g(x) as h approaches zero?

• It becomes zero.

• It becomes g'(x).

• It remains unaffected by h.

• It becomes f(x).