AP Calc AB Power Rule
34 flashcards covering AP Calc AB Power Rule for the AP-CALCULUS-AB Unit 2: Differentiation Rules section.
The Power Rule is a fundamental concept in calculus that allows you to differentiate functions of the form f(x) = x^n, where n is any real number. Defined by the College Board in the AP Calculus AB Curriculum Framework, this rule states that the derivative of x^n is n*x^(n-1). Understanding and applying the Power Rule is essential for solving problems related to rates of change and slope calculations, which are critical in both theoretical and practical applications of calculus.
On practice exams and competency assessments, you will often encounter questions that require you to differentiate polynomial functions using the Power Rule. These questions may involve straightforward applications, but be cautious of common pitfalls, such as misapplying the rule to functions that include constants or products. Additionally, students sometimes forget to simplify their final answers, which can lead to lost points. A practical tip is to always check your work by substituting values back into the original function to ensure consistency with your derivative.
Terms (34)
- 01
What is the Power Rule for differentiation?
The Power Rule states that if f(x) = x^n, then the derivative f'(x) = nx^(n-1), where n is any real number (College Board AP Course and Exam Description).
- 02
How do you apply the Power Rule to find the derivative of f(x) = 3x^4?
To apply the Power Rule, multiply the exponent by the coefficient and reduce the exponent by one: f'(x) = 43x^(4-1) = 12x^3 (College Board released AP practice exam questions).
- 03
When differentiating f(x) = -2x^5, what is the result using the Power Rule?
Using the Power Rule, the derivative is f'(x) = -10x^4, obtained by multiplying -2 by 5 and reducing the exponent by one (College Board AP Course and Exam Description).
- 04
What is the derivative of f(x) = x^0?
The derivative of f(x) = x^0, which equals 1, is f'(x) = 0, since the derivative of a constant is zero (College Board AP Course and Exam Description).
- 05
How do you differentiate f(x) = 7x^3 using the Power Rule?
To differentiate f(x) = 7x^3, apply the Power Rule: f'(x) = 37x^(3-1) = 21x^2 (College Board released AP practice exam questions).
- 06
What is the derivative of f(x) = x^(-2)?
Using the Power Rule, the derivative is f'(x) = -2x^(-3), which can also be expressed as -2/x^3 (College Board AP Course and Exam Description).
- 07
If f(x) = 2x^5 + 3x^3, what is f'(x)?
To find f'(x), differentiate each term using the Power Rule: f'(x) = 10x^4 + 9x^2 (College Board released AP practice exam questions).
- 08
When is the Power Rule applicable?
The Power Rule is applicable for functions of the form f(x) = x^n, where n is any real number (College Board AP Course and Exam Description).
- 09
What is the derivative of f(x) = 4x^7 - 5x^2 + 6?
The derivative is f'(x) = 28x^6 - 10x, applying the Power Rule to each term (College Board released AP practice exam questions).
- 10
How do you differentiate a polynomial function using the Power Rule?
Differentiate each term of the polynomial separately using the Power Rule, summing the results (College Board AP Course and Exam Description).
- 11
What is the derivative of f(x) = x^3 + 2x^2 - x + 5?
The derivative is f'(x) = 3x^2 + 4x - 1, calculated by applying the Power Rule to each term (College Board released AP practice exam questions).
- 12
What is the result of applying the Power Rule to f(x) = 10x^4?
The derivative is f'(x) = 40x^3, derived by multiplying the exponent 4 by the coefficient 10 and decreasing the exponent by one (College Board AP Course and Exam Description).
- 13
How do you find the derivative of f(x) = -x^3 + 4x^2?
Using the Power Rule, the derivative is f'(x) = -3x^2 + 8x (College Board released AP practice exam questions).
- 14
What is the derivative of f(x) = x^(1/2)?
Using the Power Rule, the derivative is f'(x) = (1/2)x^(-1/2), which can also be written as 1/(2√x) (College Board AP Course and Exam Description).
- 15
How do you apply the Power Rule to f(x) = 5x^6?
The derivative is f'(x) = 30x^5, obtained by multiplying the exponent 6 by the coefficient 5 and reducing the exponent by one (College Board released AP practice exam questions).
- 16
What is the derivative of f(x) = -3x^4 + 2x^3 - x?
The derivative is f'(x) = -12x^3 + 6x^2 - 1, calculated by applying the Power Rule to each term (College Board AP Course and Exam Description).
- 17
If f(x) = 8x^2, what is f'(x) using the Power Rule?
The derivative is f'(x) = 16x, derived by applying the Power Rule (College Board released AP practice exam questions).
- 18
What is the derivative of f(x) = x^6 - 3x^5 + 2x^4?
The derivative is f'(x) = 6x^5 - 15x^4 + 8x^3, applying the Power Rule to each term (College Board AP Course and Exam Description).
- 19
How do you differentiate f(x) = 9x^2 + 4?
Using the Power Rule, the derivative is f'(x) = 18x, as the constant term 4 has a derivative of 0 (College Board released AP practice exam questions).
- 20
What is the derivative of f(x) = 12x^3 - 7x^2 + 5x?
The derivative is f'(x) = 36x^2 - 14x + 5, calculated by applying the Power Rule to each term (College Board AP Course and Exam Description).
- 21
What happens to the derivative of a constant term in a polynomial?
The derivative of a constant term is always zero (College Board AP Course and Exam Description).
- 22
If f(x) = 2x^3 - 4x^2 + 6x - 8, what is f'(x)?
The derivative is f'(x) = 6x^2 - 8x + 6, found by applying the Power Rule to each term (College Board released AP practice exam questions).
- 23
How do you differentiate f(x) = 1/x^2 using the Power Rule?
Rewrite f(x) as x^(-2) and then apply the Power Rule: f'(x) = -2x^(-3) or -2/x^3 (College Board AP Course and Exam Description).
- 24
What is the derivative of f(x) = 3x^5 + 2x^4 - x^3?
The derivative is f'(x) = 15x^4 + 8x^3 - 3x^2, calculated by applying the Power Rule to each term (College Board released AP practice exam questions).
- 25
What is the derivative of f(x) = x^7?
Using the Power Rule, the derivative is f'(x) = 7x^6 (College Board AP Course and Exam Description).
- 26
How do you differentiate f(x) = -5x^2 + 3x + 1?
The derivative is f'(x) = -10x + 3, applying the Power Rule to each term (College Board released AP practice exam questions).
- 27
What is the derivative of f(x) = 4x^3 - 2x^2 + 1?
The derivative is f'(x) = 12x^2 - 4x, found by applying the Power Rule to each term (College Board AP Course and Exam Description).
- 28
What is the derivative of f(x) = x^8?
Using the Power Rule, the derivative is f'(x) = 8x^7 (College Board released AP practice exam questions).
- 29
If f(x) = 6x^4 - 3x^3 + 2x, what is f'(x)?
The derivative is f'(x) = 24x^3 - 9x^2 + 2, calculated by applying the Power Rule to each term (College Board AP Course and Exam Description).
- 30
How do you differentiate a constant multiplied by a power function?
The derivative of a constant multiplied by a power function follows the Power Rule: f(x) = kx^n gives f'(x) = knx^(n-1) (College Board AP Course and Exam Description).
- 31
What is the derivative of f(x) = 5x^3 + 2x^2 - 7?
The derivative is f'(x) = 15x^2 + 4x, applying the Power Rule to each term (College Board released AP practice exam questions).
- 32
What is the derivative of f(x) = 2x^4 - 3x^2 + 7x?
The derivative is f'(x) = 8x^3 - 6x + 7, found by applying the Power Rule to each term (College Board AP Course and Exam Description).
- 33
How do you differentiate f(x) = x^5 + 2x^3 - x?
The derivative is f'(x) = 5x^4 + 6x^2 - 1, calculated using the Power Rule for each term (College Board released AP practice exam questions).
- 34
What is the derivative of f(x) = 3x^2 - 4x + 1?
The derivative is f'(x) = 6x - 4, found by applying the Power Rule to each term (College Board AP Course and Exam Description).