What is the derivative of sin(2x)?

derivative of sin(2x)

We will find out the derivative of

\sin(2x)

.

Proof. Let

F(x) = \sin(2x)

f(u) = \sin(u)

and

g(x) = 2x

such that

F(x) = f(g(x))

. We are going to use the chain rule here:

\begin{align*} F'(x) = f'(g(x))g'(x). \end{align*}

We have seen here that

\frac{d}{du}\sin(u) = \cos(u)

, and

\frac{d}{dx} 2x = 2

. Therefore, we get

\begin{align*} f'(g(x)) = \cos(2x) \quad \text{and} \quad g'(x) = 2. \end{align*}

Plugging everything together, we get

\begin{align*} F'(x) &= f'(g(x))g'(x) \\ &= \cos(2x)\cdot 2\\ &= 2\cos(2x). \end{align*}

So the derivative of

\sin(2x)

2\cos(2x)