Difficulty: Introductory calculus | Prerequisites: Basic differentiation rules (power rule, sum/difference rule, constant multiple rule), unit circle values, trig identities
This topic sits at the heart of Calculus I. You already know how to differentiate polynomials. Now you learn the derivative formulas for all six trigonometric functions and apply them using the product rule and quotient rule. These formulas come back constantly in later chapters (chain rule, implicit differentiation, related rates, integration), so shaky recall here compounds quickly.
Each of the six trig functions has a derivative you need to memorise cold. Sine and cosine are the foundation; the other four (tan, cot, sec, csc) follow from quotient-rule proofs and always involve squares or products of trig functions. Most homework problems layer a trig derivative inside a product rule or quotient rule application.
Derivative of sin(x)
The rate of change of sine with respect to x. Equals cos(x). Think of it as: when sine is climbing (near x = 0), cosine is positive; when sine peaks, cosine hits zero.
Derivative of cos(x)
Equals −sin(x). The negative sign is the single most common source of sign errors in trig derivatives.
Derivative of tan(x)
Equals sec²(x). In simple terms, tangent's slope is always at least 1 (since sec²(x) ≥ 1), which is why the tangent graph steepens so quickly.
Derivative of cot(x)
Equals −csc²(x). The negative mirrors tangent's formula, and csc² mirrors sec².
Derivative of sec(x)
Equals sec(x) tan(x). A product of two trig functions, not a squared term. In simple terms, secant's rate of change depends on both how far the function is from the x-axis and the slope of the underlying tangent.
Derivative of csc(x)
Equals −csc(x) cot(x). The negative sign again: the "co-" functions always pick up a minus.
Product rule
If f(x) = u(x) · v(x), then f'(x) = u'v + uv'. Used whenever two functions are multiplied together, such as x · tan(x) or x · sin(x) · cos(x).
Quotient rule
If f(x) = u(x)/v(x), then f'(x) = (u'v − uv') / v². Used for fractions like sin(t)/(4t).
The basic six derivatives
d/dx [sin(x)] = cos(x)
d/dx [cos(x)] = −sin(x)
d/dx [tan(x)] = sec²(x)
d/dx [cot(x)] = −csc²(x)
d/dx [sec(x)] = sec(x) tan(x)
d/dx [csc(x)] = −csc(x) cot(x)
Notice the pattern: the "co-" functions (cos, cot, csc) all carry a negative sign in their derivatives. This is the single most reliable mnemonic.
Applying sum/difference and constant multiple rules
When f(x) = 10 sin(x) + 6 cos(x), differentiate term by term:
f'(x) = 10 cos(x) + 6(−sin(x)) = 10 cos(x) − 6 sin(x)
To evaluate at a point, substitute directly. For f'(−π/2):
cos(−π/2) = 0 and sin(−π/2) = −1
f'(−π/2) = 10(0) − 6(−1) = 6
The second derivative repeats the process: f''(x) = −10 sin(x) − 6 cos(x).
Differentiating sec and csc combinations
For y(x) = 5 sec(x) − 2 csc(x):
dy/dx = 5 sec(x) tan(x) − 2(−csc(x) cot(x)) = 5 sec(x) tan(x) + 2 csc(x) cot(x)
Evaluating at x = π/6 requires knowing sec(π/6) = 2/√3, tan(π/6) = 1/√3, csc(π/6) = 2, cot(π/6) = √3. Substituting and simplifying gives a numerical answer. Keep exact values until the final step.
Function | Derivative | Memory cue |
|---|---|---|
sin(x) | cos(x) | Partners: sin and cos |
cos(x) | −sin(x) | Co-function, picks up a minus |
tan(x) | sec²(x) | Squared term, always positive |
cot(x) | −csc²(x) | Co-function of tan, minus again |
sec(x) | sec(x) tan(x) | Product of itself and its partner |
csc(x) | −csc(x) cot(x) | Mirror of sec, with minus |
Product rule: (uv)' = u'v + uv'
Quotient rule: (u/v)' = (u'v − uv') / v²
Key unit circle values to have cold: sin and cos at 0, π/6, π/4, π/3, π/2, and their negatives.
Product rule example: g(s) = 4s tan(s)
Set u = 4s and v = tan(s). Then u' = 4 and v' = sec²(s).
g'(s) = u'v + uv' = 4 tan(s) + 4s sec²(s)
Evaluate at s = π/4: tan(π/4) = 1, sec(π/4) = √2, so sec²(π/4) = 2.
g'(π/4) = 4(1) + 4(π/4)(2) = 4 + 2π
Product rule with three factors: h(x) = −11x sin(x) cos(x)
You can group this as u · v where u = −11x and v = sin(x) cos(x), then apply the product rule twice (once for uv, once for sin · cos inside v). Alternatively, use the identity sin(x) cos(x) = sin(2x)/2 to simplify first. Either way, to find h'(π/2), differentiate fully, then substitute.
Using the product of three functions rule: (uvw)' = u'vw + uv'w + uvw'.
u = −11x, v = sin(x), w = cos(x)
h'(x) = −11 sin(x) cos(x) + (−11x) cos(x) cos(x) + (−11x) sin(x)(−sin(x))
h'(x) = −11 sin(x) cos(x) − 11x cos²(x) + 11x sin²(x)
At x = π/2: sin(π/2) = 1, cos(π/2) = 0.
h'(π/2) = −11(1)(0) − 11(π/2)(0) + 11(π/2)(1) = 11π/2
Quotient rule example: x(t) = sin(t) / (4t)
Set u = sin(t) and v = 4t. Then u' = cos(t) and v' = 4.
x'(t) = (cos(t) · 4t − sin(t) · 4) / (4t)² = (4t cos(t) − 4 sin(t)) / (16t²)
Simplify: x'(t) = (t cos(t) − sin(t)) / (4t²)
Simplify-first example: p(t) = (−5 tan(t) + 9) / sec(t)
Before reaching for the quotient rule, rewrite. Since 1/sec(t) = cos(t):
p(t) = (−5 tan(t) + 9) cos(t) = −5 sin(t) + 9 cos(t)
Now differentiate directly:
dp/dt = −5 cos(t) − 9 sin(t)
This is far cleaner than brute-forcing the quotient rule. Always look for simplifications involving trig identities before differentiating.
The derivatives of sin(x) cycle with period 4:
f(x) = sin(x)
f'(x) = cos(x)
f''(x) = −sin(x)
f'''(x) = −cos(x)
f⁽⁴⁾(x) = sin(x) ... and the cycle repeats
To find the nth derivative, divide n by 4 and look at the remainder:
Remainder 0: sin(x)
Remainder 1: cos(x)
Remainder 2: −sin(x)
Remainder 3: −cos(x)
For the 59th derivative of sin(x): 59 ÷ 4 = 14 remainder 3. So f⁽⁵⁹⁾(x) = −cos(x).
The same cycle applies to cos(x), shifted by one position. This pattern appears on exams regularly and takes seconds once you see it.
The tangent line to f(x) at x = a is: y = f(a) + f'(a)(x − a).
For f(x) = 4 sec(x) − 8 at the point x = π/6:
f(π/6) = 4 sec(π/6) − 8 = 4(2/√3) − 8 = 8/√3 − 8 = (8√3)/3 − 8
f'(x) = 4 sec(x) tan(x)
f'(π/6) = 4 · (2/√3) · (1/√3) = 4 · 2/3 = 8/3
So the tangent line is: y = ((8√3)/3 − 8) + (8/3)(x − π/6).
The common errors here are forgetting to evaluate f(a) (the y-coordinate of the point), or mixing up sec and cos values.
Trig derivatives describe anything that oscillates. The velocity of a mass on a spring is the derivative of its sinusoidal position function, which is why v(t) involves cos when position involves sin. In electrical engineering, the current through a capacitor is proportional to the derivative of the voltage, so a sinusoidal AC voltage produces a cosinusoidal current. Tangent-line approximations of trig functions are used in physics whenever small-angle approximations (sin θ ≈ θ) need a next-order correction.
Students often forget the negative sign in d/dx [cos(x)] = −sin(x). If your answer for a cosine derivative is positive when it should be negative, this is almost certainly why.
Students sometimes write d/dx [sec(x)] = sec²(x), confusing it with the derivative of tan(x). The derivative of sec(x) is sec(x) tan(x), a product, not a square.
When applying the quotient rule, students frequently reverse the subtraction in the numerator, writing vu' − uv' instead of u'v − uv'. The mnemonic "low d-high minus high d-low, over low squared" can help.
Students try to apply the quotient rule to expressions like (−5 tan(t) + 9)/sec(t) without first simplifying. Rewriting using identities (1/sec = cos) often makes the quotient rule unnecessary entirely.
⚠️ You will be expected to recall all six trig derivative formulas from memory. They are not typically provided on the exam.
⚠️ Product rule and quotient rule problems with trig functions are standard exam fare. Expect at least one problem combining the product rule with a trig derivative.
⚠️ The "find the nth derivative" pattern question (like the 59th derivative of sin) is a common exam question that tests whether you see the cycle. Do not try to compute 59 derivatives by hand.
⚠️ Tangent line problems require both the derivative and the function value at the point. Forgetting to compute f(a) is the most common way to lose marks.
⚠️ Evaluating trig at standard angles (π/6, π/4, π/3, π/2) is assumed knowledge. If your unit circle recall is weak, fix it before the exam.
True or false: d/dx [cos(x)] = sin(x).
False. It is −sin(x). The negative sign is essential.
Fill in the blank: d/dx [sec(x)] = ______.
sec(x) tan(x)
True or false: The 100th derivative of sin(x) is sin(x).
True. 100 ÷ 4 = 25 remainder 0, so f⁽¹⁰⁰⁾(x) = sin(x).
Fill in the blank: To find the tangent line at x = a, you need both ______ and ______.
f(a) and f'(a)
True or false: The derivative of cot(x) is csc²(x).
False. It is −csc²(x). The co-functions always carry a negative.
Q: Let f(x) = 3 sin(x) − 7 cos(x). Find f'(x).
A: f'(x) = 3 cos(x) + 7 sin(x)
Q: Find the derivative of y = sec(x) + csc(x).
A: dy/dx = sec(x) tan(x) − csc(x) cot(x)
Q: Let g(x) = x² sin(x). Find g'(x) using the product rule.
A: g'(x) = 2x sin(x) + x² cos(x)
Q: What is the 83rd derivative of cos(x)?
A: The derivatives of cos cycle: cos → −sin → −cos → sin → cos ... Divide 83 by 4: remainder 3. Starting from cos(x), position 3 gives sin(x). So f⁽⁸³⁾(x) = sin(x).
Q: Find the equation of the tangent line to y = sin(x) at x = π/3.
A: y-value: sin(π/3) = √3/2. Slope: cos(π/3) = 1/2. Tangent line: y = √3/2 + (1/2)(x − π/3).
Q: Simplify before differentiating: f(t) = sin(t)/cos(t). What is f'(t)?
A: sin(t)/cos(t) = tan(t), so f'(t) = sec²(t).
This material connects directly to the chain rule (Chapter 2.5), where you will differentiate composite trig functions like sin(3x) or cos(x²). Without the base trig derivatives memorised, the chain rule problems become unmanageable.
Trig derivatives also appear throughout integration (Chapter 4+), where you reverse the process. Knowing that the derivative of sin is cos means you will recognise that the integral of cos is sin.
Related rates problems frequently involve trigonometric functions, particularly in problems with angles (ladders against walls, rotating beacons, conical tanks viewed from above).
Trigonometric derivatives, trig differentiation, derivative of sin, derivative of cos, derivative of tan, derivative of sec, derivative of csc, derivative of cot, MTH 132, Calculus I, Michigan State, section 2.4, product rule trig, quotient rule trig, tangent line trig function, nth derivative of sine, higher order derivatives sine cosine cycle, unit circle values derivatives, sec squared, cosecant cotangent derivative