Step by Step
1
The core cycle
d/dx[sin x] = cos x. d/dx[cos x] = −sin x. d/dx[−sin x] = −cos x. d/dx[−cos x] = sin x. This four-step cycle then repeats indefinitely — differentiating sine and cosine four times in a row brings you back to where you started.
2
Why it cycles every 4
This cyclical pattern happens because sine and cosine are related through 90-degree phase shifts, and differentiating repeatedly just keeps shifting the phase by another 90 degrees, eventually returning to the original function after four steps.
3
The other four trig derivatives
d/dx[tan x] = sec²x. d/dx[cot x] = −csc²x. d/dx[sec x] = sec x·tan x. d/dx[csc x] = −csc x·cot x. These four don't follow the sin/cos cycle — they need to be memorized separately.
4
Pattern for the "co-" functions
Notice that every "co-" function (cos, cot, csc) has a negative sign in its derivative, while the non-"co-" functions (sin, tan, sec) don't. This can help you catch a sign error quickly.
Applied Walkthrough
1
Find the 10th derivative of sin(x). Since the sin/cos cycle repeats every 4 steps, divide 10 by 4: 10 = 4×2 + 2, a remainder of 2.
2
A remainder of 2 means the 10th derivative behaves like the 2nd derivative in the cycle. The cycle is: sin → cos → −sin → −cos → (repeats).
3
The 2nd derivative in that sequence is −sin(x), so the 10th derivative of sin(x) is −sin(x).
4
This same "divide by 4, use the remainder" trick works for any high-order derivative of sine or cosine, similar to the technique used for powers of i in complex numbers.
Exam Application
Exams test whether you know the full 4-step sin/cos cycle, whether you've memorized the other four trig derivatives (tan, cot, sec, csc) separately, and whether you can apply the remainder trick to find high-order derivatives.
⚠ Common Trap
The most common trap is confusing which "co-" functions get the negative sign. Remember: every "co-" function (cos, cot, csc) has a negative in its derivative; the non-"co-" functions (sin, tan, sec) don't.
✓ Quick Self-Check
1. What is the derivative of sin(x)?
cos(x).
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2. What is the derivative of cos(x)?
−sin(x).
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3. How many steps does the sin/cos derivative cycle repeat every?
4 steps.
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4. What is the derivative of tan(x)?
sec²(x).
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5. What pattern helps you remember which trig derivatives get a negative sign?
Every "co-" function (cos, cot, csc) has a negative sign in its derivative; sin, tan, and sec do not.
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