Understanding the Power Rule of Differentiation

The video tutorial provides a detailed proof of the power rule of differentiation, which states that the derivative of x raised to the power of n is n times x raised to the power of n minus 1. The proof utilizes the binomial theorem to expand expressions and evaluate combinations to determine coefficients. The tutorial walks through the process of simplifying the expression using limits, ultimately proving the power rule. This rule is a fundamental concept in differential calculus and is widely used.

What is the power rule of differentiation?

• The derivative of x^n with respect to x is n * x^(n-1).

• The derivative of x^n with respect to x is x^n.

• The derivative of x^n with respect to x is n * x^n.

• The derivative of x^n with respect to x is n^x.

What does the binomial theorem help to expand?

• Monomials

• Binomials

• Polynomials

• Trinomials

In the binomial theorem, what does 'n choose r' represent?

• The sum of n and r

• The number of ways to choose r items from n items

• The difference between n and r

• The product of n and r

What is the first step in applying the binomial theorem to f(x) = x^n?

• Subtracting x^n from the expansion

• Dividing by h

• Finding the limit as h approaches zero

• Expanding (x + h)^n using the binomial theorem

What is the value of n choose 0?

• 0

• 1

• n

• n-1

How is n choose 1 calculated?

• n factorial divided by 1 factorial

• n factorial divided by n factorial

• n factorial divided by (n-1) factorial

• n factorial divided by 2 factorial

What happens to the terms containing h as h approaches zero?

• They become infinite

• They double

• They remain constant

• They approach zero

What is the final result of the power rule proof?

• The derivative of x^n is n * x^n

• The derivative of x^n is x^n

• The derivative of x^n is n^x

• The derivative of x^n is n * x^(n-1)

Which term in the expansion does not contain a factor of h?

• All terms contain h

• The last term

• The second term

• The first term

Why is the power rule important in calculus?

• It is used only for integration

• It is rarely used

• It is the most used differentiation rule

• It is only used in advanced calculus