# 215 AP Calculus Exam Problem Power Rule

• MHB
• karush
In summary, the derivative of $y=(x^3-cos x)^5$ is $5(x^3-\cos x)^4 (3x^2 + \sin x)$ according to the application of the chain rule. Choice (E) is the closest answer, but had a sign error in the first variable factor and was missing a square in the second. It has now been updated.
karush
Gold Member
MHB
If $y=(x^3-cos x)^5$, then $y'=$(A) $\quad 5(x^3-\cos x)^4$(B) $\quad 5(3x^2+\sin x)^4$(C) $\quad 5(3x^2+\sin x)^4$(D) $\quad 5(x^3+\sin x)^4(6x+\cos x)$(E) $\quad 5(x^3+\cos x)^4(3x+\sin x)$
The Power Rule

$\displaystyle{d\over dx}x^n = nx^{n-1}$

by observation we know the derivative of the inside is multiplied to the outside
thus this leaves option (E) the other steps probably are not necessary

ok I am sure this could be worded better. but I think many students take these tests and are not used multiple choice and just plunge into timely calculations when observation could be within a few seconds

$$\displaystyle y'=5(x^3-\cos x)^4(3x^2+\sin x)$$

Last edited:
karush said:
If $y=(x^3-cos x)^5$, then $y'=$(A) $\quad 5(x^3-\cos x)^4$(B) $\quad 5(3x^2+\sin x)^4$(C) $\quad 5(3x^2+\sin x)^4$(D) $\quad 5(x^3+\sin x)^4(6x+\cos x)$(E) $\quad 5(x^3+\cos x)^4(3x+\sin x)$
The Power Rule

$\displaystyle{d\over dx}x^n = nx^{n-1}$

by observation we know the derivative of the inside is multiplied to the outside
thus this leaves option (E) the other steps probably are not necessary

ok I am sure this could be worded better. but I think many students take these tests and are not used multiple choice and just plunge into timely calculations when observation could be within a few seconds

Choice (E) is closest to being correct, but as you typed it, has a sign error in the first variable factor and is missing a square in the second.
It's not just the power rule alone, it's also an application of the chain rule.

$\dfrac{d}{dx} u^n = nu^{n-1} \cdot \dfrac{du}{dx}$

$y = (x^3-\cos{x})^5 \implies y'=5(x^3-\cos{x})^4 (3x^2 + \sin{x})$

If $y=(x^3-cos x)^5$, then $y'=$

(A) $\quad 5(x^3-\cos x)^4$

(B) $\quad 5(3x^2+\sin x)^4$

(C) $\quad 5(3x^2+\sin x)^4$

(D) $\quad 5(x^3+\sin x)^4(6x+\cos x)$

(E) $\quad 5(x^3-\cos x)^4(3x+\sin x)$
The ChainRule
$\displaystyle \dfrac{d}{dx}u^n = nu^{n-1} \cdot \dfrac{du}{dx}$
so
$y = (x^3-\cos{x})^5 \implies y'=5(x^3-\cos{x})^4 (3x^2 + \sin{x})\quad (E)$
hopefully

karush said:
If $y=(x^3-cos x)^5$, then $y'=$

(A) $\quad 5(x^3-\cos x)^4$

(B) $\quad 5(3x^2+\sin x)^4$

(C) $\quad 5(3x^2+\sin x)^4$

(D) $\quad 5(x^3+\sin x)^4(6x+\cos x)$

(E) $\quad 5(x^3-\cos x)^4(3x+\sin x)$
The ChainRule
$\displaystyle \dfrac{d}{dx}u^n = nu^{n-1} \cdot \dfrac{du}{dx}$
so
$y = (x^3-\cos{x})^5 \implies y'=5(x^3-\cos{x})^4 (3x^2 + \sin{x})\quad (E)$
hopefully

you forgot the square in the $\dfrac{du}{dx}$ factor ... again.

where ?

karush said:
where ?
Look at answer key for E). It should be a $$\displaystyle 3x^2$$, not 3x.

-Dan

got it
thanks

## 1. What is the power rule in calculus?

The power rule is a formula used to find the derivative of a function that is raised to a power. It states that for a function f(x) = x^n, the derivative is f'(x) = nx^(n-1).

## 2. How is the power rule used on the AP Calculus Exam?

The power rule is a foundational concept in calculus and is frequently tested on the AP Calculus Exam. Students may be asked to find the derivative of a function using the power rule or to apply the power rule in solving related rates or optimization problems.

## 3. Can the power rule be used for any function?

No, the power rule can only be used for functions that are in the form of x^n, where n is a constant. It cannot be used for functions that involve other variables or more complex expressions.

## 4. Are there any exceptions to the power rule?

Yes, there are a few exceptions to the power rule, such as the derivative of x^0, which is 0, and the derivative of x^1/2, which is 1/2x^(-1/2). These exceptions are important to remember when applying the power rule.

## 5. How can I practice using the power rule for the AP Calculus Exam?

There are many resources available for practicing the power rule, including past AP Calculus exams, review books, and online practice problems. It is important to not only understand the formula but also to be able to apply it in various contexts, so practicing with different types of problems is key.

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