Differentiate

cos
⁡
(
3
𝑥
)

Question

Differentiate

cos
⁡
(
3
𝑥
)

BlackTom AI · Accepted Answer

In solving differentiation problems, I’ll analyze each choice carefully to see whether it matches the correct application of the chain rule.

Option 1: '3 sin(3x)'. This option has the correct sinusoidal function sin(3x) but the sign is positive. Differentiating cos(3x) should involve a negative sign due to the derivative of cos being -sin, so a positive 3 sin(3x) is incorrect.

Option 2: '−3 sin(3x)'. This aligns with the chain rule: if y = cos(u) with u = 3x, then dy/du = -sin(u) and du/dx = 3. Multiplying gives dy/dx = (-sin(3x)) * 3 = −3 sin(3x). This is the correct result, since the negative sign comes from the derivative of cosine.

Option 3: '−3 sin(x)'. Here the inner function is treated as x rather than 3x, which neglects the inner derivative of u = 3x. This mismatch means the result is incorrect because it ignores the rate of change inside the argument of the sine.

Option 4: '3 cos(3x)'. This option keeps a cosine function instead of sine, and also misses the essential negative sign from differentiating cosine. It does not reflect the chain rule application for cos(3x) and is therefore wrong.

Overall, the key error patterns are sign mistakes, incorrect inner argument handling, and using the wrong trigonometric function. The only option that correctly applies the chain rule to cos(3x) is the one yielding −3 sin(3x).

Differentiate cos ⁡ ( 3 𝑥 )

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