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AP Calculus AB: 4.1.2 A Quick Proof of the Power Rule

Mathematics12 CardsCreated 3 months ago

This content explains why the power rule for differentiation is valid, particularly for integer exponents, using the binomial theorem as a foundational proof. It emphasizes the need for mathematical proof beyond observed patterns and provides examples applying the rule to various functions.

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quick proof of power rule

  • In math, it is not enough to find patterns. Once you find one, it is necessary to prove that it holds in general. To prove the power rule for integer exponents, use the binomial theorem to express the general case.

  • The power rule states that if N is a rational number, then the function is differentiable and Nx^N-1.

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Key Terms

Term
Definition

quick proof of power rule

  • In math, it is not enough to find patterns. Once you find one, it is necessary to prove that it holds in general. To prove the power rule f...

note

  • If the power rule is true, finding derivatives will be much easier and quicker.

  • But how do you know that the rule is true fo...

Suppose a particle’s position is given by f(t)=t^4, where t is measured in seconds and f(t) is given in centimeters. At what time is the velocity of the particle equal to 116?

t=1/4

Suppose f(x)=x^k. What is f′(x)?

f′(x)=kx^k−1

Find the derivative of f (x) = x 11.

f′(x)=11x^10

Suppose f(x)=x^1/2. What is f′(x)?

f′(x)=√x/2x

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AP Calculus AB: 4.1.2 A Quick Proof of the Power Rule
TermDefinition

quick proof of power rule

  • In math, it is not enough to find patterns. Once you find one, it is necessary to prove that it holds in general. To prove the power rule for integer exponents, use the binomial theorem to express the general case.

  • The power rule states that if N is a rational number, then the function is differentiable and Nx^N-1.

note

  • If the power rule is true, finding derivatives will be much easier and quicker.

  • But how do you know that the rule is true for all cases? It is not enough to see that it holds sometimes. You must prove that the rule works under given conditions.

  • Use the binomial theorem to prove the power rule for integer powers. The binomial theorem illustrates how a binomial expands when raised to any given power.

  • Consider the function x^N.

  • Notice that the binomial term raised to a power creates many additional terms that must be considered. However, the binomial theorem shows that only the first two terms are important, since all other terms will have a factor of delta x^2.

  • Taking the limit after canceling the delta x-terms eliminates all of the extra terms generated by expanding the binomial.

  • Notice that the binomial theorem only holds for integers, so this proof only works for integer powers.

Suppose a particle’s position is given by f(t)=t^4, where t is measured in seconds and f(t) is given in centimeters. At what time is the velocity of the particle equal to 116?

t=1/4

Suppose f(x)=x^k. What is f′(x)?

f′(x)=kx^k−1

Find the derivative of f (x) = x 11.

f′(x)=11x^10

Suppose f(x)=x^1/2. What is f′(x)?

f′(x)=√x/2x

Find the derivative of f (x) = x ^9.

9x^8

Find the derivative of f (x) = x ^8.

8x^7

Find the derivative of f (x) = x ^2.

2x

Suppose f(x)=x^6.5872. What is f′(x)?

f′(x)=6.5872x^5.5872

Suppose f(x)=x^0.1. What is f′(x)?

f′(x)=0.1x^−0.9

Suppose f(x)=x^3/2. What is f′(x)?

f′(x)=3/2x1/2