Derivative of sin(x)
The derivative of the sine function is one of the first results every calculus student memorizes — and one of the few that comes out perfectly clean.
Step-by-step solution
You are differentiating f(x) = sin(x), a bare sine with the variable itself as its argument — no inner function, so no chain rule is needed.
Sine's derivative is defined directly from the limit of the difference quotient: limh→0 [sin(x+h) − sin(x)] / h = cos(x). This limit relies on the fact that sin(h)/h → 1 as h → 0.
f′(x) = cos(x). Done — no simplification required.
Why it works
A useful way to see why this is true without the limit algebra: picture a point moving around the unit circle. Its height is sin(x) and its horizontal position is cos(x). The rate at which the height changes is exactly the horizontal coordinate — when the point is at the far right of the circle (height 0, moving straight up), the height changes fastest, and cos(0) = 1 agrees.
The derivatives of sine and cosine chain together in a four-step cycle: sin → cos → −sin → −cos → sin. Knowing that cycle lets you write down any higher derivative of sin(x) instantly — the 4th derivative returns to sin(x), the 99th derivative is the same as the 3rd, and so on. Remember that this clean result only holds when x is measured in radians; in degrees a factor of π/180 appears.
Common mistakes
- Writing the derivative as −cos(x). The minus sign belongs to the derivative of cosine, not sine — mixing up which of the pair picks up the negative is the single most common trig-derivative error.
- Applying the result to sin(2x) or sin(x²) without the chain rule. The clean answer cos(x) only holds when the argument is exactly x.
- Working in degrees. All the standard derivative formulas assume radians.
Practice problems
Find the derivative of 4 sin(x)
Answer: 4 cos(x) — constants ride along untouched.
Find the second derivative of sin(x)
Answer: −sin(x) — differentiate cos(x) to get −sin(x).
Find the slope of sin(x) at x = 0
Answer: cos(0) = 1, which is why sin(x) ≈ x near zero.