Derivative of `sin(2*x)`

Step-by-step differentiation

Answer

\frac{d}{d x} [sin (2 x)]

2 cos (2 x)

Step-by-step solution

1Chain Rule — sin
$\frac{d}{d x} [sin (2 x)] = cos (2 x) \cdot \frac{d}{d x} [2 x]$
This is $sin$ applied to $u = 2 x$ . The outer derivative is $cos (u)$ , which we multiply by $u^{'}$ (chain rule).
↳Find u′: Differentiate the inner function, $u = 2 x$ :
2Constant Multiple Rule
$\frac{d}{d x} [2 x] = 2 \cdot \frac{d}{d x} [x]$
The factor $2$ is constant — pull it outside so only the variable part needs differentiating.
3Power Rule
$\frac{d}{d x} [x] = 1$
$x$ is $x^{1}$ : by the power rule, the exponent $1$ comes down and the power drops to $x^{0} = 1$ , so the derivative is $1$ .
=Combine
$\frac{d}{d x} [2 x] = 2 \cdot 1 = 2$
Multiply: $2 \cdot 1 = 2$ .
=Combine
$\frac{d}{d x} [sin (2 x)] = cos (2 x) \cdot 2 = 2 cos (2 x)$
With $u^{'} = 2$ , multiply by the outer derivative to complete the chain rule.
✓Final Answer
$\frac{d}{d x} [sin (2 x)] = 2 cos (2 x)$
The first derivative simplifies to $2 cos (2 x)$ .

Understanding this derivative

Differentiating sin(2x) exercises 4 distinct techniques — the chain rule, the power rule, the trigonometric derivatives and the constant multiple rule. Problems like this one are useful practice precisely because the rules have to be combined in the right order rather than applied in isolation.

The chain rule handles the composite structure here: differentiate the outer function while leaving the inner expression alone, then multiply by the derivative of that inner expression. Forgetting that final multiplication is one of the most common mistakes in differentiation. The power rule is the workhorse step: bring the exponent down as a coefficient and reduce the exponent by one. The trigonometric derivatives follow the standard cycle: sine differentiates to cosine, cosine to negative sine, and tangent to secant squared. The sign on the cosine derivative is a frequent source of errors, so it is worth double-checking. Constant coefficients pass straight through differentiation — they are carried along unchanged while the variable part is differentiated.

A good way to verify a derivative by hand is to compare it against the step-by-step breakdown above — each step names the rule being applied, so you can pinpoint exactly where your own work diverges if the answers differ.

Learn the rules used here

Basic Rules Derivatives of Transcendental Functions Mastering the Chain Rule

Try a different expression

Open Calculator

Derivative of sin(2*x)

Step-by-step solution

Understanding this derivative

Learn the rules used here

Derivative of `sin(2*x)`