Calc 1 Power Rule
37 flashcards covering Calc 1 Power Rule for the CALCULUS-1 Calc 1 Topics section.
The Power Rule is a fundamental concept in Calculus I that provides a straightforward method for differentiating polynomial functions. According to the curriculum set by the College Board for AP Calculus, the Power Rule states that if f(x) = x^n, then the derivative f'(x) = n*x^(n-1). Understanding this rule is essential for students as it forms the basis for more complex differentiation techniques encountered in higher-level calculus.
On practice exams and competency assessments, questions related to the Power Rule often require students to differentiate various polynomial expressions. Common traps include misapplying the rule by forgetting to decrease the exponent correctly or failing to simplify the result. Additionally, students may overlook the importance of applying the rule in combination with other differentiation techniques, such as the product or quotient rules. A practical tip to remember is to always double-check that the exponent is reduced by one after applying the Power Rule, as this is a frequent source of errors.
Terms (37)
- 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 a real number (Stewart Calculus, differentiation chapter).
- 02
How do you apply the Power Rule to find the derivative of x^5?
To find the derivative of x^5 using the Power Rule, multiply the exponent by the coefficient (1) and subtract 1 from the exponent: f'(x) = 5x^(5-1) = 5x^4 (Stewart Calculus, differentiation chapter).
- 03
What is the derivative of x^0?
The derivative of x^0, which equals 1, is 0, since the Power Rule gives f'(x) = 0x^(0-1) = 0 (Stewart Calculus, differentiation chapter).
- 04
Using the Power Rule, what is the derivative of 3x^4?
The derivative of 3x^4 is found by applying the Power Rule: f'(x) = 34x^(4-1) = 12x^3 (Stewart Calculus, differentiation chapter).
- 05
What is the derivative of x^(-2)?
Using the Power Rule, the derivative of x^(-2) is f'(x) = -2x^(-2-1) = -2/x^3 (Stewart Calculus, differentiation chapter).
- 06
How do you differentiate a polynomial using the Power Rule?
To differentiate a polynomial, apply the Power Rule to each term separately. For example, for f(x) = 2x^3 + 5x^2 - x, the derivative is f'(x) = 6x^2 + 10x - 1 (Stewart Calculus, differentiation chapter).
- 07
What is the derivative of 7x^3 - 4x^2 + 2?
Using the Power Rule, the derivative is f'(x) = 21x^2 - 8x + 0, simplifying to 21x^2 - 8x (Stewart Calculus, differentiation chapter).
- 08
When applying the Power Rule, what happens to the constant multiplier?
When applying the Power Rule, the constant multiplier remains unchanged and is multiplied by the derivative of the variable term (Stewart Calculus, differentiation chapter).
- 09
What is the derivative of 5x^7 + 3x^2 - 8?
The derivative is f'(x) = 35x^6 + 6x - 0, simplifying to 35x^6 + 6x (Stewart Calculus, differentiation chapter).
- 10
How do you find the derivative of a function like f(x) = 4x^5 + 2x^3 - 7x + 1?
Apply the Power Rule to each term: f'(x) = 20x^4 + 6x^2 - 7 (Stewart Calculus, differentiation chapter).
- 11
What is the derivative of x^1/2?
Using the Power Rule, the derivative of x^(1/2) is f'(x) = (1/2)x^(-1/2) = 1/(2√x) (Stewart Calculus, differentiation chapter).
- 12
How is the Power Rule used in finding the derivative of a function involving negative exponents?
The Power Rule applies to negative exponents as well; for example, the derivative of x^(-3) is f'(x) = -3x^(-4) = -3/x^4 (Stewart Calculus, differentiation chapter).
- 13
What is the derivative of 2x^3 - 5x^2 + 4x - 1?
The derivative is f'(x) = 6x^2 - 10x + 4 (Stewart Calculus, differentiation chapter).
- 14
How do you differentiate a term with a fractional exponent like x^(3/2)?
Using the Power Rule, the derivative of x^(3/2) is f'(x) = (3/2)x^(3/2 - 1) = (3/2)x^(1/2) = (3√x)/2 (Stewart Calculus, differentiation chapter).
- 15
What is the derivative of a constant function, such as f(x) = 5?
The derivative of a constant function is always 0, as constants do not change (Stewart Calculus, differentiation chapter).
- 16
How do you differentiate the function f(x) = 6x^4 + 2x - 9?
The derivative is f'(x) = 24x^3 + 2 (Stewart Calculus, differentiation chapter).
- 17
What is the derivative of x^6?
Using the Power Rule, the derivative of x^6 is f'(x) = 6x^(6-1) = 6x^5 (Stewart Calculus, differentiation chapter).
- 18
How does the Power Rule apply to the function f(x) = -2x^3 + x^2?
Applying the Power Rule, the derivative is f'(x) = -6x^2 + 2x (Stewart Calculus, differentiation chapter).
- 19
What is the derivative of the function f(x) = 4x^7 - 3x^5 + 2x?
The derivative is f'(x) = 28x^6 - 15x^4 + 2 (Stewart Calculus, differentiation chapter).
- 20
How do you find the derivative of f(x) = 2x^4 + 3x^3 - x?
Using the Power Rule, the derivative is f'(x) = 8x^3 + 9x^2 - 1 (Stewart Calculus, differentiation chapter).
- 21
What is the derivative of x^8?
The derivative of x^8 is f'(x) = 8x^(8-1) = 8x^7 (Stewart Calculus, differentiation chapter).
- 22
How do you differentiate a function with multiple terms using the Power Rule?
Differentiate each term separately using the Power Rule; for example, for f(x) = x^3 + x^2 + x, f'(x) = 3x^2 + 2x + 1 (Stewart Calculus, differentiation chapter).
- 23
What is the derivative of 10x^2 - 4x + 7?
The derivative is f'(x) = 20x - 4 (Stewart Calculus, differentiation chapter).
- 24
How do you apply the Power Rule to a term like -x^4?
The derivative of -x^4 is f'(x) = -4x^(4-1) = -4x^3 (Stewart Calculus, differentiation chapter).
- 25
What is the derivative of 0.5x^3?
Using the Power Rule, the derivative is f'(x) = 0.5 3x^(3-1) = 1.5x^2 (Stewart Calculus, differentiation chapter).
- 26
What is the derivative of x^(1/3)?
Using the Power Rule, the derivative is f'(x) = (1/3)x^(-2/3) = 1/(3√(x^2)) (Stewart Calculus, differentiation chapter).
- 27
How do you differentiate f(x) = 7x^2 + 4x^3 - 5?
The derivative is f'(x) = 14x + 12x^2 (Stewart Calculus, differentiation chapter).
- 28
What is the derivative of x^9?
The derivative of x^9 is f'(x) = 9x^(9-1) = 9x^8 (Stewart Calculus, differentiation chapter).
- 29
How do you find the derivative of a function like f(x) = 3x^5 - 2x^2 + 1?
Using the Power Rule, the derivative is f'(x) = 15x^4 - 4x (Stewart Calculus, differentiation chapter).
- 30
What is the derivative of -3x^4 + 2x^3?
The derivative is f'(x) = -12x^3 + 6x^2 (Stewart Calculus, differentiation chapter).
- 31
How do you differentiate a term like 8x^6?
The derivative is f'(x) = 48x^5 using the Power Rule (Stewart Calculus, differentiation chapter).
- 32
What is the derivative of x^10?
The derivative of x^10 is f'(x) = 10x^(10-1) = 10x^9 (Stewart Calculus, differentiation chapter).
- 33
How do you find the derivative of a function with a negative exponent like f(x) = x^(-1)?
Using the Power Rule, the derivative is f'(x) = -1x^(-2) = -1/x^2 (Stewart Calculus, differentiation chapter).
- 34
What is the derivative of 2x^5?
The derivative is f'(x) = 10x^4 using the Power Rule (Stewart Calculus, differentiation chapter).
- 35
How do you differentiate a function with multiple terms like f(x) = x^3 + 2x^2 - x?
Differentiate each term: f'(x) = 3x^2 + 4x - 1 (Stewart Calculus, differentiation chapter).
- 36
What is the derivative of 4x^2 - 3x + 5?
The derivative is f'(x) = 8x - 3 (Stewart Calculus, differentiation chapter).
- 37
What is the derivative of 6x^3?
The derivative is f'(x) = 18x^2 using the Power Rule (Stewart Calculus, differentiation chapter).