Questions: Derivatives of Trigonometric Functions

Question 1

What is d/dx[cos(2x)]?

• A. sin(2x)
• B. -sin(2x)
• C. 2sin(2x)
• D. -2sin(2x)

Answer: (D)

Two errors combine here: forgetting the negative sign (d/dx[cos(x)] = -sin(x), not sin(x)) and applying or omitting the chain rule. The chain rule requires multiplying by the derivative of the inner function 2x, which is 2. So d/dx[cos(2x)] = -sin(2x) · 2 = -2sin(2x). Option A forgets both the negative and the chain rule. Option B forgets only the chain rule. Option C forgets only the negative sign.

Question 2

The formula d/dx[sin(x)] = cos(x) is established by:

• A. Observing empirically that the sine and cosine graphs seem related
• B. Applying the limit definition of the derivative and the squeeze theorem result lim(h→0) sin(h)/h = 1
• C. Applying the chain rule to the unit circle parametrization
• D. Noting that the second derivative of sin(x) is -sin(x), so the first must be cos(x)

Answer: (B)

The derivation applies the limit definition (sin(x+h) - sin(x))/h, expands using the angle addition identity sin(x+h) = sin(x)cos(h) + cos(x)sin(h), and uses two limits established by the squeeze theorem: lim sin(h)/h = 1 and lim (cos(h)-1)/h = 0. The result follows rigorously from these limit facts. The other options are informal or circular — option D in particular assumes you already know the answer.

Question 3

All four secondary trig derivatives (tan, cot, sec, csc) can be derived from the sine and cosine derivatives using only the quotient rule and Pythagorean identities.

• T. True
• F. False

Answer: True

This is exactly how the derivations work. For example, tan(x) = sin(x)/cos(x), so the quotient rule gives (cos²(x) + sin²(x))/cos²(x), and the Pythagorean identity sin²(x) + cos²(x) = 1 simplifies this to 1/cos²(x) = sec²(x). The same pattern applies to cot, sec, and csc — each is expressed as a ratio of sin and cos, the quotient rule is applied, and a Pythagorean identity compresses the numerator.

Question 4

d/dx[sec(x)] = sec(x)cot(x)

• T. True
• F. False

Answer: False

The correct derivative is d/dx[sec(x)] = sec(x)tan(x), not sec(x)cot(x). Since sec(x) = 1/cos(x), the quotient rule gives sin(x)/cos²(x) = (1/cos(x)) · (sin(x)/cos(x)) = sec(x)tan(x). A useful mnemonic: sec pairs with tan, and csc pairs with -csc·cot (the co-functions form the negated pair). Mixing them up — putting cot with sec — is the most common error.

Question 5

Why must you apply the chain rule when differentiating sin(3x), but the formula d/dx[sin(x)] = cos(x) alone is sufficient for sin(x)?

Model answer: sin(x) has the identity function x as its argument, whose derivative is 1, so the chain rule multiplier is 1 and can be ignored. In sin(3x), the argument is the function 3x, whose derivative is 3. The chain rule says: differentiate the outer function (keeping the inner function untouched), then multiply by the derivative of the inner function. So d/dx[sin(3x)] = cos(3x) · 3 = 3cos(3x).

The six trig derivative formulas assume the argument is simply x. Whenever the argument is any other expression — 3x, x², x²+1 — the chain rule adds an extra multiplicative factor equal to the derivative of that expression. Failure to apply the chain rule is the most common error in trig differentiation practice.