Calc 1 Inverse Trig Derivatives
35 flashcards covering Calc 1 Inverse Trig Derivatives for the CALCULUS-1 Calc 1 Topics section.
Inverse trigonometric derivatives are a crucial topic in Calculus I, focusing on the differentiation of inverse trigonometric functions such as arcsin, arccos, and arctan. This topic is defined by the curriculum standards set forth by the College Board for AP Calculus, which outlines the fundamental concepts and skills students must master in single-variable calculus courses.
In practice exams and competency assessments, questions regarding inverse trig derivatives often require students to apply the chain rule and recognize the specific derivatives of these functions. A common pitfall is confusing the derivatives of inverse functions with their corresponding trigonometric functions, leading to incorrect answers. Additionally, students may overlook the importance of including the absolute value in the domain restrictions for certain derivatives, which can cause errors in their final solutions.
Remember to double-check your work to ensure that you include any necessary domain restrictions when presenting your answers.
Terms (35)
- 01
What is the derivative of arcsin(x)?
The derivative of arcsin(x) is 1/√(1 - x²), valid for -1 < x < 1 (Stewart Calculus, derivative rules chapter).
- 02
What is the derivative of arccos(x)?
The derivative of arccos(x) is -1/√(1 - x²), valid for -1 < x < 1 (Stewart Calculus, derivative rules chapter).
- 03
What is the derivative of arctan(x)?
The derivative of arctan(x) is 1/(1 + x²), valid for all x (Stewart Calculus, derivative rules chapter).
- 04
What is the derivative of arccot(x)?
The derivative of arccot(x) is -1/(1 + x²), valid for all x (Stewart Calculus, derivative rules chapter).
- 05
What is the derivative of arcsec(x)?
The derivative of arcsec(x) is 1/(|x|√(x² - 1)), valid for |x| > 1 (Stewart Calculus, derivative rules chapter).
- 06
What is the derivative of arccsc(x)?
The derivative of arccsc(x) is -1/(|x|√(x² - 1)), valid for |x| > 1 (Stewart Calculus, derivative rules chapter).
- 07
When differentiating arcsin(x), what is the restriction on x?
The restriction on x for differentiating arcsin(x) is -1 < x < 1 (Stewart Calculus, derivative rules chapter).
- 08
What is the first step in finding the derivative of arctan(2x)?
The first step is to apply the chain rule, differentiating the outer function arctan(u) where u = 2x (Stewart Calculus, derivative rules chapter).
- 09
How do you find the second derivative of arcsin(x)?
To find the second derivative of arcsin(x), differentiate 1/√(1 - x²) using the quotient rule or chain rule (Stewart Calculus, derivative rules chapter).
- 10
What is the derivative of arctan(3x + 1)?
The derivative is 3/(1 + (3x + 1)²), applying the chain rule (Stewart Calculus, derivative rules chapter).
- 11
When differentiating arccos(5x), what rule is applied?
The chain rule is applied, giving the derivative as -5/(√(1 - (5x)²)) (Stewart Calculus, derivative rules chapter).
- 12
What is the derivative of arcsec(2x)?
The derivative of arcsec(2x) is 2/(|2x|√((2x)² - 1)), valid for |2x| > 1 (Stewart Calculus, derivative rules chapter).
- 13
What is the derivative of arccsc(4x)?
The derivative of arccsc(4x) is -4/(|4x|√((4x)² - 1)), valid for |4x| > 1 (Stewart Calculus, derivative rules chapter).
- 14
How do you find the derivative of a composition involving arctan?
Use the chain rule to differentiate the outer arctan function and then multiply by the derivative of the inner function (Stewart Calculus, derivative rules chapter).
- 15
What is the derivative of arcsin(3x²)?
The derivative is (6x)/(√(1 - (3x²)²)), applying the chain rule (Stewart Calculus, derivative rules chapter).
- 16
What is the derivative of arccos(2x + 1)?
The derivative is -2/(√(1 - (2x + 1)²)), using the chain rule (Stewart Calculus, derivative rules chapter).
- 17
What is the derivative of arctan(x² + 1)?
The derivative is (2x)/(1 + (x² + 1)²), applying the chain rule (Stewart Calculus, derivative rules chapter).
- 18
What is the derivative of arccot(5x)?
The derivative of arccot(5x) is -5/(1 + (5x)²), using the chain rule (Stewart Calculus, derivative rules chapter).
- 19
What is the derivative of arcsec(x²)?
The derivative is (2x)/(|x²|√(x⁴ - 1)), valid for |x²| > 1 (Stewart Calculus, derivative rules chapter).
- 20
What is the derivative of arccsc(x + 1)?
The derivative is -1/(|x + 1|√((x + 1)² - 1)), valid for |x + 1| > 1 (Stewart Calculus, derivative rules chapter).
- 21
When is the derivative of arcsin(x) undefined?
The derivative of arcsin(x) is undefined at x = -1 and x = 1 (Stewart Calculus, derivative rules chapter).
- 22
What is the derivative of arctan(u) where u = 1/x?
The derivative is -1/(x² + 1), applying the chain rule (Stewart Calculus, derivative rules chapter).
- 23
What is the derivative of arccos(3x)?
The derivative is -3/(√(1 - (3x)²)), using the chain rule (Stewart Calculus, derivative rules chapter).
- 24
What is the derivative of arcsin(2x + 3)?
The derivative is 2/(√(1 - (2x + 3)²)), applying the chain rule (Stewart Calculus, derivative rules chapter).
- 25
How do you differentiate arccos(4x² + 1)?
Use the chain rule to find the derivative as -8x/(√(1 - (4x² + 1)²)) (Stewart Calculus, derivative rules chapter).
- 26
What is the derivative of arctan(1/x)?
The derivative is -1/(x² + 1), applying the chain rule (Stewart Calculus, derivative rules chapter).
- 27
What is the derivative of arccot(2x + 1)?
The derivative is -2/(1 + (2x + 1)²), using the chain rule (Stewart Calculus, derivative rules chapter).
- 28
What is the derivative of arcsec(3x)?
The derivative is 3/(|3x|√((3x)² - 1)), valid for |3x| > 1 (Stewart Calculus, derivative rules chapter).
- 29
What is the derivative of arccsc(2x)?
The derivative is -2/(|2x|√((2x)² - 1)), valid for |2x| > 1 (Stewart Calculus, derivative rules chapter).
- 30
What is the derivative of arcsin(5x)?
The derivative is 5/(√(1 - (5x)²)), using the chain rule (Stewart Calculus, derivative rules chapter).
- 31
What is the derivative of arccos(x² + 2)?
The derivative is -2x/(√(1 - (x² + 2)²)), using the chain rule (Stewart Calculus, derivative rules chapter).
- 32
What is the derivative of arctan(2x + 3)?
The derivative is 2/(1 + (2x + 3)²), applying the chain rule (Stewart Calculus, derivative rules chapter).
- 33
What is the derivative of arccot(4x)?
The derivative is -4/(1 + (4x)²), using the chain rule (Stewart Calculus, derivative rules chapter).
- 34
What is the derivative of arcsec(5x)?
The derivative is 5/(|5x|√((5x)² - 1)), valid for |5x| > 1 (Stewart Calculus, derivative rules chapter).
- 35
What is the derivative of arccsc(3x + 1)?
The derivative is -3/(|3x + 1|√((3x + 1)² - 1)), valid for |3x + 1| > 1 (Stewart Calculus, derivative rules chapter).