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

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

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

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

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

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

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 $f{\left(x \right)} = \cos{\left(x \right)}$:

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

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

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

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