What is the Derivative of sin x?

d/dx(sin x) = cos x

This means: if f(x) = sin x, then f'(x) = cos x.

Key observations:

• The derivative of sin x is cos x (a 90° phase shift).
• The derivative of cos x is −sin x.
• Differentiating sin x twice gives −sin x, and four times gives back sin x.
• This cyclical pattern: sin x → cos x → −sin x → −cos x → sin x repeats every 4 derivatives.

Using the definition of derivative: d/dx(sin x) = lim(h→0) [sin(x+h) − sin(x)] / h

Step 1: Expand sin(x+h) using the addition formula: sin(x+h) = sin x · cos h + cos x · sin h

Step 2: Substitute: = lim(h→0) [sin x · cos h + cos x · sin h − sin x] / h = lim(h→0) [sin x(cos h − 1) + cos x · sin h] / h

Step 3: Split the limit: = sin x · lim(h→0) [(cos h − 1)/h] + cos x · lim(h→0) [sin h / h]

Step 4: Apply the two standard limits:

• lim(h→0) [sin h / h] = 1 (fundamental trigonometric limit)
• lim(h→0) [(cos h − 1)/h] = 0 (derived from the first limit)

Step 5: Substitute: = sin x · 0 + cos x · 1 = cos x

Therefore, d/dx(sin x) = cos x. Q.E.D.

All six standard trigonometric derivatives:

1. d/dx(sin x) = cos x
2. d/dx(cos x) = −sin x
3. d/dx(tan x) = sec²x
4. d/dx(cot x) = −cosec²x
5. d/dx(sec x) = sec x · tan x
6. d/dx(cosec x) = −cosec x · cot x

Memory tip:

• The derivatives of 'co-' functions (cos, cot, cosec) all have a negative sign.
• sin ↔ cos (swap and negate the derivative of cosine)
• tan ↔ sec² (square the secant)
• cot ↔ −cosec² (square the cosecant, negate)

When sin is applied to a function of x, use the chain rule: d/dx[sin(f(x))] = cos(f(x)) · f'(x)

Examples:

1. d/dx(sin 2x) = cos 2x · 2 = 2 cos 2x
2. d/dx(sin x²) = cos(x²) · 2x = 2x cos(x²)
3. d/dx(sin(3x+1)) = cos(3x+1) · 3 = 3 cos(3x+1)
4. d/dx(sin√x) = cos√x · (1/2√x) = cos√x / (2√x)
5. d/dx(sin²x) = 2 sin x · cos x = sin 2x [using chain rule: 2 sin x · d/dx(sin x)]

General rule: 'Differentiate the outer function, keep the inner, multiply by derivative of the inner.'

The derivatives of sin x follow a repeating cycle of period 4:

• 1st derivative: d/dx(sin x) = cos x
• 2nd derivative: d²/dx²(sin x) = d/dx(cos x) = −sin x
• 3rd derivative: d³/dx³(sin x) = d/dx(−sin x) = −cos x
• 4th derivative: d⁴/dx⁴(sin x) = d/dx(−cos x) = sin x
• 5th derivative: d⁵/dx⁵(sin x) = cos x (cycle repeats)

General formula for the nth derivative:

• If n ≡ 1 (mod 4): cos x
• If n ≡ 2 (mod 4): −sin x
• If n ≡ 3 (mod 4): −cos x
• If n ≡ 0 (mod 4): sin x

This cyclic property is significant in solving differential equations and Fourier analysis.