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Second derivative of $$$\sin{\left(2 x \right)}$$$

The calculator will find the second derivative of $$$\sin{\left(2 x \right)}$$$, with steps shown.

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Your Input

Find $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(2 x \right)}\right)$$$.

Solution

Find the first derivative $$$\frac{d}{dx} \left(\sin{\left(2 x \right)}\right)$$$

The function $$$\sin{\left(2 x \right)}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ and $$$g{\left(x \right)} = 2 x$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(2 x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(2 x\right)\right)}$$

The derivative of the sine is $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(2 x\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(2 x\right)$$

Return to the old variable:

$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(2 x\right) = \cos{\left({\color{red}\left(2 x\right)} \right)} \frac{d}{dx} \left(2 x\right)$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 2$$$ and $$$f{\left(x \right)} = x$$$:

$$\cos{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(2 x\right)\right)} = \cos{\left(2 x \right)} {\color{red}\left(2 \frac{d}{dx} \left(x\right)\right)}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$2 \cos{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 2 \cos{\left(2 x \right)} {\color{red}\left(1\right)}$$

Thus, $$$\frac{d}{dx} \left(\sin{\left(2 x \right)}\right) = 2 \cos{\left(2 x \right)}$$$.

Next, $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(2 x \right)}\right) = \frac{d}{dx} \left(2 \cos{\left(2 x \right)}\right)$$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 2$$$ and $$$f{\left(x \right)} = \cos{\left(2 x \right)}$$$:

$${\color{red}\left(\frac{d}{dx} \left(2 \cos{\left(2 x \right)}\right)\right)} = {\color{red}\left(2 \frac{d}{dx} \left(\cos{\left(2 x \right)}\right)\right)}$$

The function $$$\cos{\left(2 x \right)}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ and $$$g{\left(x \right)} = 2 x$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$$2 {\color{red}\left(\frac{d}{dx} \left(\cos{\left(2 x \right)}\right)\right)} = 2 {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dx} \left(2 x\right)\right)}$$

The derivative of the cosine is $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:

$$2 {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dx} \left(2 x\right) = 2 {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dx} \left(2 x\right)$$

Return to the old variable:

$$- 2 \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(2 x\right) = - 2 \sin{\left({\color{red}\left(2 x\right)} \right)} \frac{d}{dx} \left(2 x\right)$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 2$$$ and $$$f{\left(x \right)} = x$$$:

$$- 2 \sin{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(2 x\right)\right)} = - 2 \sin{\left(2 x \right)} {\color{red}\left(2 \frac{d}{dx} \left(x\right)\right)}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$- 4 \sin{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = - 4 \sin{\left(2 x \right)} {\color{red}\left(1\right)}$$

Thus, $$$\frac{d}{dx} \left(2 \cos{\left(2 x \right)}\right) = - 4 \sin{\left(2 x \right)}$$$.

Therefore, $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(2 x \right)}\right) = - 4 \sin{\left(2 x \right)}$$$.

Answer

$$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(2 x \right)}\right) = - 4 \sin{\left(2 x \right)}$$$A