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

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

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

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

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

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

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

Apply the power rule $\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$ with $n = 2$:

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

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

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

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

Simplify:

$$- 2 \sin{\left(x \right)} \cos{\left(x \right)} = - \sin{\left(2 x \right)}$$

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

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

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 $c = -1$ and $f{\left(x \right)} = \sin{\left(2 x \right)}$:

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

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 $\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$:

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

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

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

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

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