Derivatives of Trigonometric Functions

The Six Basic Trig Derivatives

• d/dx[sin x] = cos x
• d/dx[cos x] = −sin x
• d/dx[tan x] = sec²x
• d/dx[csc x] = −csc x cot x
• d/dx[sec x] = sec x tan x
• d/dx[cot x] = −csc²x

The first two are foundational. The others follow from the Quotient Rule applied to tan, csc, sec, cot.

The Sine Derivative Cycle

Differentiate sin(x) four times and you return to sin(x): sin → cos → −sin → −cos → sin. This cycle repeats with period 4. It means: the nth derivative of sin(x) follows the pattern: n≡0: sin, n≡1: cos, n≡2: −sin, n≡3: −cos.

Proof That d/dx[sin x] = cos x

Using the limit definition and two key limits (lim sin(h)/h = 1 and lim (cos h−1)/h = 0): d/dx[sin x] = lim[(sin(x+h)−sin(x))/h] = lim[sin(x)cos(h)+cos(x)sin(h)−sin(x))/h] = sin(x)·0 + cos(x)·1 = cos(x).

Chain Rule with Trig Functions

• d/dx[sin(3x²)] = cos(3x²)·6x
• d/dx[cos(eˣ)] = −sin(eˣ)·eˣ
• d/dx[tan(√x)] = sec²(√x)·(1/2x^(−1/2)) = sec²(√x)/(2√x)

Inverse Trig Derivatives

• d/dx[arcsin x] = 1/√(1−x²)
• d/dx[arccos x] = −1/√(1−x²)
• d/dx[arctan x] = 1/(1+x²)
• These are derived using implicit differentiation on the inverse relationship.

Why Trigonometric Derivatives Matter

Trigonometric functions are the mathematical language of periodicity — anything that oscillates, rotates, or cycles is described by sines and cosines. Sound waves, light waves, electrical current, planetary orbits, pendulums, tides, and heartbeats are all modelled trigonometrically. Their derivatives describe the rates at which these phenomena change — velocity of a pendulum, rate of current change in an AC circuit, angular acceleration of a spinning body. Mastering trig derivatives is not optional if you study physics, engineering, or any quantitative science.

Deriving d/dx[sin x] Rigorously

Using the limit definition and the sine addition formula sin(A+B) = sin A cos B + cos A sin B:

d/dx[sin x] = lim(h→0) [sin(x+h) − sin x] / h = lim(h→0) [sin x cos h + cos x sin h − sin x] / h

= lim(h→0) sin x · [(cos h − 1)/h] + cos x · [sin h / h]

= sin x · 0 + cos x · 1 = cos x

The two limit results used — lim(h→0) sin(h)/h = 1 and lim(h→0)(cos h − 1)/h = 0 — were proved using the Squeeze Theorem and geometric arguments on the unit circle. The entire chain of reasoning is watertight.

The Full Table — All Six Derivatives

• d/dx[sin x] = cos x
• d/dx[cos x] = −sin x
• d/dx[tan x] = sec²x (from Quotient Rule on sin x/cos x)
• d/dx[cot x] = −csc²x (from Quotient Rule on cos x/sin x)
• d/dx[sec x] = sec x · tan x (from Quotient Rule on 1/cos x)
• d/dx[csc x] = −csc x · cot x (from Quotient Rule on 1/sin x)

The Four-Step Sine Cycle

Differentiating sin(x) repeatedly cycles through four functions: sin → cos → −sin → −cos → sin → ... The period is 4. This means the 100th derivative of sin(x) is sin(x) (since 100 = 25×4, we return to the start). The 101st derivative is cos(x). This cyclic property makes computing arbitrarily high derivatives of trig functions trivial once you know the pattern.

Chain Rule Applications — Detailed

Inverse Trigonometric Derivatives — Derived

These are obtained by implicit differentiation. For y = arcsin(x): sin y = x. Differentiating implicitly: cos y · (dy/dx) = 1, so dy/dx = 1/cos y. Since sin y = x, cos y = √(1−x²) (taking positive root for the principal value). Therefore d/dx[arcsin x] = 1/√(1−x²). Similarly:

• d/dx[arccos x] = −1/√(1−x²)
• d/dx[arctan x] = 1/(1+x²)
• d/dx[arccot x] = −1/(1+x²)
• d/dx[arcsec x] = 1/(|x|√(x²−1))
• d/dx[arccsc x] = −1/(|x|√(x²−1))

Typical Exam Questions

• Find d/dx[x² sin x] — requires Product Rule, no Chain Rule.
• Find d/dx[sin²(3x)] — requires Chain Rule twice (power outer, sin middle, 3x inner).
• Find d/dx[arctan(x²+1)] — inverse trig outer, polynomial inner.
• If f(x) = cos(x), find f⁽⁵⁰⁾(x) — use the cycle: 50 mod 4 = 2, so answer is −cos x.