Derivative of sin(x) is cos(x)

Answer

\mathrm{cos}\left(x\right)

Derivative of sin(x)

Rule Used: Trigonometric Rule
Steps: Recognize the trigonometric function $\sin(x)$ . These six derivatives are standard; a cofunction (cos, cot, csc) picks up a minus sign.
Apply the standard derivative: $\frac{d}{dx} \sin(x)$ = $\cos(x)$ .
Final answer: $\cos(x)$

The derivative of sin(x) is cos(x), provided x is measured in radians.

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Derivative of sin(x) = cos(x) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-sin-x

Derivative

sin(x) → cos(x)engine: cos(x)

Graph of the function and its derivative

The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.

Why the derivative has this form

Radians are not a cosmetic choice here. The special limit sin(h)/h -> 1 holds only when h is a pure number, which is what an angle in radians is. In degrees the same limit picks up a factor of pi/180, and every trigonometric derivative would carry an extra constant. This is the main reason calculus uses radians.

The proof starts from the addition formula sin(x+h) = sin(x)cos(h) + cos(x)sin(h). Subtracting sin(x) and dividing by h splits the quotient into two pieces: sin(x)·(cos(h)-1)/h and cos(x)·sin(h)/h. The first tends to 0 and the second tends to cos(x), so the whole expression tends to cos(x). No graph reading is required.

The result matches the shape of the sine wave. Where sin(x) climbs most steeply, near x = 0, cos(x) is close to 1. At the peaks x = pi/2 and x = -pi/2 the slope is 0 and cos(x) vanishes. Where sine falls, cosine is negative. Differentiating once more gives -sin(x), which is why the second derivative of sine is the original function with a minus sign.

Proof from the addition formula

Write the difference quotient (sin(x+h) - sin(x))/h.
Expand sin(x+h) = sin(x)cos(h) + cos(x)sin(h).
Subtract sin(x) and group: sin(x)(cos(h)-1)/h + cos(x)sin(h)/h.
Use the first special limit: (cos(h)-1)/h -> 0, so the sin(x) term vanishes.
Use the second special limit: sin(h)/h -> 1, so the cos(x) term becomes cos(x).
Conclude d/dx sin(x) = cos(x). Both limits require h in radians.
Check at x = 0: cos(0) = 1, the steepest positive slope of the sine curve.

Worked example

Slope of sin(x) at x = pi/3

The derivative at any point is cos(x), so evaluate cos(pi/3).
cos(pi/3) = 1/2. The sine curve is rising at half the steepness it has at the origin.
The point itself is (pi/3, sin(pi/3)) = (pi/3, √3/2), about (1.047, 0.866).
The tangent line is y - √3/2 = (1/2)(x - pi/3). A secant to pi/3 + 0.001 has a slope near 0.5, not 0.866.

Answer: 1/2

Common mistakes

Answering -cos(x)

A minus sign appears when differentiating cos(x), whose derivative is -sin(x), or on the second derivative of sin(x). Sine itself has a positive slope near the origin, so its derivative there must be positive.

Using degrees

With x in degrees, d/dx sin(x) = (pi/180)cos(x). The extra factor comes from sin(h)/h -> pi/180 for h measured in degrees. Stating the radian convention is part of stating the derivative.

Confusing sin(x)^2 with sin(x^2)

This page covers sin(x) alone. sin(x)^2 needs the chain rule on the outer square and gives 2 sin(x)cos(x); sin(x^2) needs it on the inner x^2 and gives 2x cos(x^2). Neither is cos(x).

Derivative of sin(2x) — 2cos(2x). Chain rule on the same outer sine function.
Trigonometric derivatives — The rule page lists this example and the others in the same family.
Derivatives of trigonometric functions — Same family of differentiation, different function.
Derivative of cos(x) — Same family of differentiation, different function.
Derivative of tan(x) — Same family of differentiation, different function.
Derivative of sec(x) — Same family of differentiation, different function.
Derivative of csc(x) — Same family of differentiation, different function.
Derivative of cot(x) — Same family of differentiation, different function.