17 Lecture

What is the derivative of the sine function?

a. cosine function

b. tangent function

c. cosecant function

d. secant function

Answer: a. cosine function

What is the derivative of the cosine function?

a. sine function

b. tangent function

c. cosecant function

d. negative sine function

Answer: d. negative sine function

What is the derivative of the tangent function?

a. cosine function

b. cosecant function

c. square of the secant function

d. negative square of the cosecant function

Answer: c. square of the secant function

What is the derivative of the cotangent function?

a. sine function

b. cosine function

c. negative square of the cosecant function

d. negative square of the secant function

Answer: c. negative square of the cosecant function

What is the derivative of the secant function?

a. cosecant function

b. tangent function

c. product of the secant and tangent functions

d. negative product of the secant and tangent functions

Answer: c. product of the secant and tangent functions

What is the derivative of the cosecant function?

a. secant function

b. cotangent function

c. negative product of the cosecant and cotangent functions

d. product of the cosecant and cotangent functions

Answer: c. negative product of the cosecant and cotangent functions

What is the derivative of sin(x) + cos(x)?

a. cos(x) - sin(x)

b. sin(x) + cos(x)

c. sin(x) - cos(x)

d. cos(x) + sin(x)

Answer: a. cos(x) - sin(x)

What is the derivative of tan(x) * sec(x)?

a. sec^2(x)

b. sec(x) * tan(x)

c. sec(x) + tan(x)

d. tan^2(x)

Answer: b. sec(x) * tan(x)

What is the derivative of cos(2x)?

a. -2sin(2x)

b. -sin(2x)

c. 2sin(2x)

d. -2cos(2x)

Answer: d. -2sin(2x)

What is the derivative of arcsin(x)?

a. 1/sqrt(1-x^2)

b. -1/sqrt(1-x^2)

c. 1/(1-x^2)

d. -1/(1-x^2)

Answer: a. 1/sqrt(1-x^2)

What is the derivative of the sine function?

Answer: The derivative of the sine function is the cosine function.

What is the derivative of the cosine function?

Answer: The derivative of the cosine function is the negative of the sine function.

What is the derivative of the tangent function?

Answer: The derivative of the tangent function is the square of the secant function.

What is the derivative of the cotangent function?

Answer: The derivative of the cotangent function is the negative of the square of the cosecant function.

What is the derivative of the secant function?

Answer: The derivative of the secant function is the product of the secant and tangent functions.

What is the derivative of the cosecant function?

Answer: The derivative of the cosecant function is the negative of the product of the cosecant and cotangent functions.

How do you find the derivative of a product of two trigonometric functions?

Answer: You can use the product rule to find the derivative of a product of two trigonometric functions.

How do you find the derivative of a sum of two trigonometric functions?

Answer: You can use the sum rule to find the derivative of a sum of two trigonometric functions.

How do you find the derivative of the inverse trigonometric functions?

Answer: You can use the chain rule to find the derivative of the inverse trigonometric functions.

How do you find the derivative of a composition of a trigonometric function and another function?

Answer: You can use the chain rule to find the derivative of a composition of a trigonometric function and another function.

Trigonometric functions play an important role in calculus, and the derivatives of these functions are crucial to solving many problems. In this article, we will explore the derivatives of trigonometric functions and how to apply them to solve problems in calculus.

The derivatives of trigonometric functions are as follows:

Derivative of sine function: The derivative of the sine function is the cosine function. That is, if y = sin(x), then dy/dx = cos(x). Derivative of cosine function: The derivative of the cosine function is the negative of the sine function. That is, if y = cos(x), then dy/dx = -sin(x). Derivative of tangent function: The derivative of the tangent function is the square of the secant function. That is, if y = tan(x), then dy/dx = sec^2(x). Derivative of cotangent function: The derivative of the cotangent function is the negative of the square of the cosecant function. That is, if y = cot(x), then dy/dx = -csc^2(x). Derivative of secant function: The derivative of the secant function is the product of the secant and tangent functions. That is, if y = sec(x), then dy/dx = sec(x)tan(x). Derivative of cosecant function: The derivative of the cosecant function is the negative of the product of the cosecant and cotangent functions. That is, if y = csc(x), then dy/dx = -csc(x)cot(x). To find the derivative of a function, we need to use differentiation rules such as the product rule, quotient rule, or chain rule. Let's look at some examples of finding the derivatives of trigonometric functions using these rules.

Example 1: Find the derivative of y = cos(x)sin(x).

Solution: We can use the product rule to find the derivative of this function. Applying the product rule, we get: dy/dx = cos(x)cos(x) - sin(x)sin(x) = cos^2(x) - sin^2(x) Therefore, the derivative of y = cos(x)sin(x) is dy/dx = cos^2(x) - sin^2(x).

Example 2: Find the derivative of y = 2x + sin(x).

Solution: We can use the sum rule to find the derivative of this function. Applying the sum rule, we get: dy/dx = d/dx(2x) + d/dx(sin(x)) = 2 + cos(x) Therefore, the derivative of y = 2x + sin(x) is dy/dx = 2 + cos(x).

Example 3: Find the derivative of y = tan(x).

Solution: We can use the quotient rule to find the derivative of this function. Applying the quotient rule, we get: dy/dx = [d/dx(sin(x))] / [cos^2(x)] = cos(x) / cos^2(x) = sec(x) Therefore, the derivative of y = tan(x) is dy/dx = sec(x).

Example 4: Find the derivative of y = csc(x).

Solution: We can use the chain rule to find the derivative of this function. Applying the chain rule, we get: dy/dx = d/dx(1/sin(x)) = -1/sin^2(x) * d/dx(sin(x)) = -cot(x)csc(x) Therefore, the derivative of y = csc(x) is dy/dx = -cot(x)csc(x).

Example 5: Find the derivative of y = sec(x)tan(x).