# Question #e058f

Nov 9, 2017

Use that dang ol' chain rule...

#### Explanation:

...but you have to apply it more than once...

$\frac{d}{\mathrm{dx}} \left({\cos}^{2} \left(\pi \cdot x\right)\right)$

$= 2 \cos \left(\pi \cdot x\right) \cdot \frac{d}{\mathrm{dx}} \left(\cos \left(\pi \cdot x\right)\right)$

$= 2 \cos \left(\pi \cdot x\right) \cdot \left(- \sin \left(\pi \cdot x\right) \cdot \frac{d}{\mathrm{dx}} \left(\pi \cdot x\right)\right)$

$= 2 \cos \left(\pi \cdot x\right) \cdot \pi \left(- \sin \left(\pi \cdot x\right)\right)$

$= - 2 \pi \left(\cos \left(\pi \cdot x\right) \cdot \sin \left(\pi \cdot x\right)\right)$

(Wolfram reminds me that you can use the double angle identity - $2 \cos \left(a\right) \sin \left(a\right) = \sin \left(2 a\right)$, so you can rewrite the above as:

$- \pi \left(\sin \left(2 \pi \cdot x\right)\right)$

GOOD LUCK