Trigonometric derivatives

Question 1

What is the derivative of sin(x)?

Answer 1

cos(x)

This is a fundamental result in calculus related to trigonometric functions.

Question 2

What is the derivative of cos(x)?

Answer 2

−sin(x)

This indicates that the slope of the cosine function is negative sine.

Question 3

What is the derivative of tan(x)?

Answer 3

sec²(x)

The secant squared function is the derivative of the tangent function.

Question 4

What is the derivative of csc(x)?

Answer 4

−csc(x)cot(x)

This shows the relationship between the cosecant and cotangent functions.

Question 5

What is the derivative of sec(x)?

Answer 5

sec(x)tan(x)

The derivative involves both secant and tangent functions.

Question 6

What is the derivative of cot(x)?

Answer 6

−csc²(x)

This indicates that the derivative of cotangent is negative cosecant squared.

Question 7

What is the derivative of sin⁻¹(x)?

Answer 7

1/√(1 - x²)

This is the derivative of the inverse sine function.

Question 8

What is the derivative of cos⁻¹(x)?

Answer 8

−1/√(1 - x²)

This represents the rate of change of the inverse cosine function.

Question 9

What is the derivative of tan⁻¹(x)?

Answer 9

1/(1 + x²)

This shows the derivative of the inverse tangent function.

Question 10

What is the derivative of csc⁻¹(x)?

Answer 10

−1/|x|√(x² - 1)

This indicates the derivative of the inverse cosecant function.

Question 11

What is the derivative of sec⁻¹(x)?

Answer 11

1/|x|√(x² - 1)

This shows how the inverse secant function changes.

Question 12

What is the derivative of cot⁻¹(x)?

Answer 12

−1/(1 + x²)

This indicates the rate of change of the inverse cotangent function.