# How do you use the chain rule to differentiate cos(10csc10x)?

$100 \sin \left(10 \csc \left(10 x\right)\right) \csc \left(10 x\right) \cot \left(10 x\right)$
$\frac{d}{\mathrm{dx}} \left(\cos \left(10 \csc \left(10 x\right)\right)\right) = - \sin \left(10 \csc \left(10 x\right)\right) \cdot \frac{d}{\mathrm{dx}} \left(10 \csc \left(10 x\right)\right)$ (chain rule)
$= - \sin \left(10 \csc \left(10 x\right)\right) \cdot 10 \left(- \csc \left(10 x\right) \cot \left(10 x\right)\right) \cdot \frac{d}{\mathrm{dx}} \left(10 x\right)$ (chain rule again)
$= 10 \sin \left(10 \csc \left(10 x\right)\right) \cdot \left(\csc \left(10 x\right) \cot \left(10 x\right)\right) \cdot 10$
$= 100 \sin \left(10 \csc \left(10 x\right)\right) \csc \left(10 x\right) \cot \left(10 x\right)$