Solving Problems On Differentiation Of Trigonometric Functions

 

Rules of Differentiation for Trigonometric Functions

A comprehensive guide with formulas, objective questions, and step-by-step solutions.

1. Derivative of Sine [d/dx(sin x)]

The derivative of the sine function is the cosine function.

d/dx(sin x) = cos x

Objective Question 1: Find the derivative of f(x) = sin(3x).

• A) cos(3x)
• B) 3cos(3x)
• C) -3cos(3x)
• D) 3sin(3x)

Correct Answer: B Solution:
Using the Chain Rule:
Let u = 3x, so du/dx = 3.
d/dx(sin u) = cos u * du/dx
d/dx(sin(3x)) = cos(3x) * 3 = 3cos(3x)

2. Derivative of Cosine [d/dx(cos x)]

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

d/dx(cos x) = -sin x

Objective Question 2: Differentiate y = cos(x^2) with respect to x.

• A) -sin(x^2)
• B) 2x+sin(x^2)
• C) -2x*sin(x^2)
• D) -2x*cos(x^2)

Correct Answer: C Solution:
Apply the Chain Rule:
The derivative of the outer function cos(u) is -sin(u).
The derivative of the inner function x^2 is 2x.
Multiplying them together gives: -2x*sin(x^2).

3. Derivative of Tangent [d/dx(tan x)]

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

d/dx(tan x) = sec^2 x

Objective Question 3: What is d/dx(5tan x)?

• A) 5sec^2 x
• B) sec^2(5x)
• C) 5cot x
• D) -5sec^2 x

Correct Answer: A Solution:
By the constant multiple rule, pull out the constant 5:
d/dx(5tan x) = 5 * d/dx(tan x) = 5sec^2 x.

4. Derivative of Cosecant [d/dx(csc x)]

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

d/dx(csc x) = -csc x * cot x

Objective Question 4: Find the derivative of f(x) = csc(4x).

• A) -csc(4x)cot(4x)
• B) 4csc(4x)cot(4x)
• C) -4csc(4x)cot(4x)
• D) -4sec(4x)tan(4x)

Correct Answer: C Solution:
Using the Chain Rule:
Differentiate the outer function: -csc(4x)cot(4x).
Multiply by the derivative of the inner function (4x), which is 4.
Result: -4csc(4x)cot(4x).

5. Derivative of Secant [d/dx(sec x)]

The derivative of the secant function is the product of secant and tangent.

d/dx(sec x) = sec x * tan x

Objective Question 5: Evaluate the derivative of y = x*sec x.

• A) sec x * tan x
• B) sec x * (1 + x*tan x)
• C) sec x + tan x
• D) x*sec x*tan x

Correct Answer: B Solution:
Apply the Product Rule [d/dx(uv) = u'v + uv']:
Let u = x implies u' = 1.
Let v = sec x implies v' = sec x * tan x.
dy/dx = (1)(sec x) + (x)(sec x * tan x) = sec x + x*sec x*tan x.
Factoring out sec x gives: sec x * (1 + x*tan x).

6. Derivative of Cotangent [d/dx(cot x)]

The derivative of the cotangent function is the negative cosecant squared function.

d/dx(cot x) = -csc^2 x

Objective Question 6: If y = cot x, find dy/dx at x = pi/4.

• A) -1
• B) -2
• C) 1
• D) 2

Correct Answer: B Solution:
First, find the general derivative: dy/dx = -csc^2 x.
Recall that csc x = 1/sin x. At x = pi/4, sin(pi/4) = 1/sqrt(2).
Therefore, csc(pi/4) = sqrt(2).
Squaring it gives [sqrt(2)]^2 = 2.
Substituting this back into our formula yields: -2.