Find $$$\frac{d^{3}}{dx^{3}} \left(\sin{\left(x \right)}\right)$$$

The calculator will find $$$\frac{d^{3}}{dx^{3}} \left(\sin{\left(x \right)}\right)$$$, with steps shown.

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Find $$$\frac{d^{3}}{dx^{3}} \left(\sin{\left(x \right)}\right)$$$.


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

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

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

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

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

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

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

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

Next, $$$\frac{d^{3}}{dx^{3}} \left(\sin{\left(x \right)}\right) = \frac{d}{dx} \left(- \sin{\left(x \right)}\right)$$$

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 $$$c = -1$$$ and $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:

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

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

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

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

Therefore, $$$\frac{d^{3}}{dx^{3}} \left(\sin{\left(x \right)}\right) = - \cos{\left(x \right)}$$$.


$$$\frac{d^{3}}{dx^{3}} \left(\sin{\left(x \right)}\right) = - \cos{\left(x \right)}$$$A