Derivative of Sine

Q: What is the derivative of Sine?

The derivative of $\sin(x)$ is $\cos(x)$. This is one of the fundamental derivatives in calculus that you should memorize.

Q: Is the derivative of Sine always the same?

Yes, the derivative formula for $\sin(x)$ is constant — it does not depend on x. However, when composed with inner functions (e.g., $\sin(x)$ of u(x)), the chain rule applies.

Complete guide with formula, proof, examples, and graph.

Quick Answer

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

The derivative of \sin(x) is:

Proof / Derivation

Step-by-step derivation of the derivative formula.

Apply the limit definition of the derivative.

\frac{d}{dx}\sin(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h}

Expand using the sine addition formula: sin(x+h) = sin(x)cos(h) + cos(x)sin(h).

= \lim_{h \to 0} \frac{\sin(x)\cos(h) + \cos(x)\sin(h) - \sin(x)}{h}

Group terms and factor out sin(x) and cos(x).

= \lim_{h \to 0} \left[ \sin(x)\frac{\cos(h)-1}{h} + \cos(x)\frac{\sin(h)}{h} \right]

Evaluate the well-known limits: lim (cos(h)-1)/h = 0 and lim sin(h)/h = 1.

= \sin(x) \cdot 0 + \cos(x) \cdot 1 = \cos(x)

Graph

Visualization of Sine and its derivative.

f(x) = \sin(x)

f(x) = \sin(x)

f'(x) = \cos(x)

f'(x) = \cos(x)

Domain:

(-\infty, +\infty)

Range:

[-1, 1]

Worked Examples

Step-by-step solutions using the chain rule and other techniques.

Find: $\frac{d}{dx}\sin(3x)$

Solution: $3\cos(3x)$

Apply chain rule: \frac{d}{dx}\sin(u) = \cos(u) \cdot u'

Let u = 3x, so u' = 3

\frac{d}{dx}\sin(3x) = \cos(3x) \cdot 3 = 3\cos(3x)

Find: $\frac{d}{dx}\sin^2(x)$

Solution: $2\sin(x)\cos(x) = \sin(2x)$

Apply chain rule: \frac{d}{dx}[u]^2 = 2u \cdot u'

Let u = \sin(x), so u' = \cos(x)

\frac{d}{dx}\sin^2(x) = 2\sin(x) \cdot \cos(x) = \sin(2x)

Calculate Any Derivative

Use our free online derivative calculator to verify your answers or solve more complex functions.

Open Derivative Calculator

Related Derivatives

\cos(x)

\tan(x)

\cot(x)

\sec(x)

\csc(x)

\ln(x)

\log_a(x)

\sqrt{x}

\arcsin(x)

Frequently Asked Questions

What is the derivative of Sine?

The derivative of

\sin(x)

\cos(x)

. This is one of the fundamental derivatives in calculus that you should memorize.

How do you prove the derivative of Sine?

The proof uses the limit definition of the derivative. See the Proof section above for the complete step-by-step derivation.

Is the derivative of Sine always the same?

Yes, the derivative formula for

\sin(x)

is constant — it does not depend on x. However, when composed with inner functions (e.g.,

\sin(x)

of u(x)), the chain rule applies.

Where is the derivative of Sine undefined?

The derivative is undefined where the original function is not differentiable. Check the domain section for details.

Why is the derivative of Sine important?

This derivative appears frequently in physics (wave motion), engineering (signal processing), economics (oscillating models), and many other fields involving periodic or growth phenomena.