What is the derivative of sin x?

The derivative of \sin x is \cos x. This is one of the fundamental trigonometric derivatives in calculus, and it comes from applying the formal limit definition of the derivative combined with trigonometric identities.

Why is the derivative of sin x equal to cos x?

This result comes from the limit definition of the derivative and trigonometric properties. When you work through \lim_{h \to 0} \frac{\sin(x + h) - \sin x}{h} using the sine addition formula and limit algebra, you end up with \cos x. Geometrically, the slope of the sine curve at any point equals the height of the cosine curve at that same point.

How do you find the derivative of sin(2x)?

You use the chain rule combined with the sine derivative rule. Differentiate the outside function (sine) to get cosine, then multiply by the derivative of the inside function (2x). This gives: \frac{d}{dx}[\sin(2x)] = \cos(2x) \cdot 2 = 2\cos(2x).

Is the derivative of sin x the same as the derivative of cos x?

No, they are different. The derivative of \sin x is \cos x, but the derivative of \cos x is -\sin x (note the negative sign). A common mistake is to forget this negative sign on the cosine derivative, so keep these two rules clearly separated.

Derivative of Sin x

The derivative of sin x is cos x, giving the slope of the sine curve; use this rule when differentiating any sine function.

\frac{d}{dx}\sin x = \cos x

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What each symbol means

What Derivative of Sin x takes

Symbol	Meaning
$x$	The independent variable, measured in radians (not degrees) — the input to the sine function. If $x$ is in degrees instead of radians, this rule does not apply and you must convert or use a different form of the derivative rule.

When to use it

Use this rule whenever you need to find the instantaneous rate of change of a sine function.

Level

Usually taught in: Calculus I

Worked examples

1. Find the derivative of sin x and evaluate it at x = 0

Problem

Find the derivative of

f(x) = \sin x

and then evaluate

f'(0)

$f(x) = \sin x$
We are given the function $f(x) = \sin x$ and need to find its derivative.
$f'(x) = \cos x$
By the sine derivative rule, the derivative of $\sin x$ is $\cos x$ .
$f'(0) = \cos(0) = 1$
We evaluate the derivative at $x = 0$ by substituting into $\cos x$ , which gives $\cos(0) = 1$ .

Answer: $f'(x) = \cos x$

This example demonstrates both finding the general derivative formula and evaluating it at a specific point. The derivative $\cos x$ represents the instantaneous slope of the sine curve at every point, and at $x = 0$ (where the sine curve crosses zero with steepest positive slope), that slope is exactly 1.

2. Differentiate a combination of sine and a linear term

Problem

Find

\frac{d}{dx}[3\sin x - 2x]

$\frac{d}{dx}[3\sin x - 2x]$
We need to find the derivative of the entire expression, which contains two separate terms.
$= 3\frac{d}{dx}[\sin x] - 2\frac{d}{dx}[x]$
We apply the constant multiple rule and sum/difference rule to handle each term separately: the derivative of $3\sin x$ is $3$ times the derivative of $\sin x$ , and the derivative of $2x$ is $2$ times the derivative of $x$ .
$= 3\cos x - 2$
We apply the sine derivative rule to get $\cos x$ and the power rule to get $1$ , giving us $3\cos x - 2 \cdot 1 = 3\cos x - 2$ .

Answer: $3\cos x - 2$

This problem requires combining the sine derivative rule with other derivative rules. The constant multiple rule and sum rule allow us to handle each term independently before applying the appropriate derivative formula to each one.

3. Find the rate of change of daily revenue modeled by sine

Problem

A coffee shop's daily revenue is modeled by

R(t) = 300 + 150\sin(t)

dollars, where

t

is the number of days since opening. Find

\frac{dR}{dt}

, the rate of change of revenue with respect to time.

$R(t) = 300 + 150\sin(t)$
We are given the revenue function: a constant baseline of $300$ plus a sinusoidal oscillation term $150\sin(t)$ .
$\frac{dR}{dt} = \frac{d}{dt}[300] + \frac{d}{dt}[150\sin(t)]$
We apply the sum rule to split the derivative into two separate parts: the derivative of the constant $300$ and the derivative of the term $150\sin(t)$ .
$= 0 + 150\cos(t) = 150\cos(t)$
The derivative of a constant is $0$ , and the derivative of $150\sin(t)$ is $150$ times the derivative of $\sin(t)$ , which is $\cos(t)$ .

Answer: $\frac{dR}{dt} = 150\cos(t) \text{ dollars per day}$

In this real-world context, the derivative tells us how fast the revenue is changing at each moment. The sine term creates a periodic fluctuation around the $300 baseline, and the derivative $150\cos(t)$ oscillates between $+150$ and $-150$ dollars per day, showing when revenue is increasing and decreasing most rapidly.

Common mistakes

Where Derivative of Sin x usually goes wrong

The mistake	Why it is wrong	The fix
Writing $\frac{d}{dx}[\sin x] = -\cos x$ with a negative sign	Students often confuse the sine derivative with the cosine derivative, which does include a negative sign: $\frac{d}{dx}[\cos x] = -\sin x$ .	Remember: $\frac{d}{dx}[\sin x] = \cos x$ (positive), and $\frac{d}{dx}[\cos x] = -\sin x$ (negative). Write these two rules side by side to keep them straight.
For $\frac{d}{dx}[\sin(3x)]$ , writing $\cos(3x)$ instead of $3\cos(3x)$	Students apply the sine rule to get $\cos(3x)$ but forget to use the chain rule and multiply by the derivative of the inside function, which is $3$ .	When differentiating $\sin(f(x))$ for any function $f(x)$ , use the chain rule: $\frac{d}{dx}[\sin(f(x))] = \cos(f(x)) \cdot f'(x)$ . For $\sin(3x)$ , multiply: $\cos(3x) \cdot 3 = 3\cos(3x)$ .
For $\frac{d}{dx}[x \sin x]$ , writing $\cos x$ instead of $\sin x + x\cos x$	Students see sine and apply only the sine derivative rule, but fail to recognize that $x$ and $\sin x$ are multiplied together and the product rule must be used.	Use the product rule: $\frac{d}{dx}[x \sin x] = 1 \cdot \sin x + x \cdot \cos x = \sin x + x\cos x$ . The sine derivative rule must be combined with the product rule for multiplied expressions.

Tips and when to use something else

Remember that $\frac{d}{dx}[\sin x] = \cos x$ but $\frac{d}{dx}[\cos x] = -\sin x$ — note the negative sign on the cosine derivative.
Whenever the argument of sine is not exactly $x$ (like $\sin(2x)$ or $\sin(x^2)$ ), use the Chain Rule to multiply by the derivative of the inside function.
If $x$ and sine are multiplied together, such as $x\sin x$ or $x^2\sin x$ , use the Product Rule, not just the sine derivative rule.
The sine derivative rule applies only when sine's argument is exactly $x$ ; all other cases require combining it with other differentiation rules.

Related formulas

Definition of the DerivativeThe Definition of the Derivative finds the instantaneous rate of change at a point using limits; the foundation for all derivative calculations.Chain RuleThe Chain Rule tells you how to differentiate a composite function—when one function is plugged into another—by multiplying the outer and inner derivatives.Derivative of Cos xThe derivative of cosine x equals negative sine x, the fundamental rule for finding rates of change of oscillating cosine functions.Higher Order DerivativesHigher order derivatives describe how the rate of change is changing; use them to find concavity and acceleration in functions.

Frequently asked questions

What is the derivative of sin x?: The derivative of $\sin x$ is $\cos x$ . This is one of the fundamental trigonometric derivatives in calculus, and it comes from applying the formal limit definition of the derivative combined with trigonometric identities.
Why is the derivative of sin x equal to cos x?: This result comes from the limit definition of the derivative and trigonometric properties. When you work through $\lim_{h \to 0} \frac{\sin(x + h) - \sin x}{h}$ using the sine addition formula and limit algebra, you end up with $\cos x$ . Geometrically, the slope of the sine curve at any point equals the height of the cosine curve at that same point.
How do you find the derivative of sin(2x)?: You use the chain rule combined with the sine derivative rule. Differentiate the outside function (sine) to get cosine, then multiply by the derivative of the inside function ( $2x$ ). This gives: $\frac{d}{dx}[\sin(2x)] = \cos(2x) \cdot 2 = 2\cos(2x)$ .
Is the derivative of sin x the same as the derivative of cos x?: No, they are different. The derivative of $\sin x$ is $\cos x$ , but the derivative of $\cos x$ is $-\sin x$ (note the negative sign). A common mistake is to forget this negative sign on the cosine derivative, so keep these two rules clearly separated.

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Reviewed 2026-09-18