Derivative of sin(2*x)

Step-by-step solution

1. 1Chain Rule — sin
   $$\frac{d}{dx}\left[\sin \left(2x\right)\right]=\cos \left(2x\right)\cdot \frac{d}{dx}\left[2x\right]$$

   This is $\sin$ applied to $u=2x$. The outer derivative is $\cos (u)$, which we multiply by $u^{\prime }$ (chain rule).

2. ↳Find u′: Differentiate the inner function, $u=2x$:
3. 2Constant Multiple Rule
   $$\frac{d}{dx}\left[2x\right]=2\cdot \frac{d}{dx}\left[x\right]$$

   The factor $2$ is constant — pull it outside so only the variable part needs differentiating.

4. 3Variable Rule
   $$\frac{d}{dx}\left[x\right]=1$$

   Differentiating $x$ with respect to itself always gives $1$.

5. =Combine
   $$\frac{d}{dx}\left[2x\right]=2\cdot 1=2$$

   Multiply: $2\cdot 1=2$.

6. =Combine
   $$\frac{d}{dx}\left[\sin \left(2x\right)\right]=\cos \left(2x\right)\cdot 2=2\cos \left(2x\right)$$

   With $u^{\prime }=2$, multiply by the outer derivative to complete the chain rule.

7. ✓Final Answer
   $$\frac{d}{dx}\left[\sin \left(2x\right)\right]=2\cos \left(2x\right)$$

   The first derivative simplifies to $2\cos \left(2x\right)$.