What is the derivative of y = cos(cos(cos(x)))?

Oct 9, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \sin \left(\cos \left(\cos \left(x\right)\right)\right) \sin \left(\cos x\right) \sin \left(x\right)$

Explanation:

In order to differentiate a function of a function, say $y , = f \left(g \left(x\right)\right)$, where we have to find $\frac{\mathrm{dy}}{\mathrm{dx}}$, we need to do (a) substitute $u = g \left(x\right)$, which gives us $y = f \left(u\right)$. Then we need to use a formula called Chain Rule, which states that $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \times \frac{\mathrm{du}}{\mathrm{dx}}$. In fact if we have something like $y = f \left(g \left(h \left(x\right)\right)\right)$, we can have $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{df}} \times \frac{\mathrm{df}}{\mathrm{dg}} \times \frac{\mathrm{dg}}{\mathrm{dh}}$

Hence for $y = \cos \left(\cos \left(\cos \left(x\right)\right)\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \sin \left(\cos \left(\cos \left(x\right)\right)\right) \times \frac{d}{\mathrm{dx}} \left(\cos \left(\cos \left(x\right)\right)\right)$

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

= $- \sin \left(\cos \left(\cos \left(x\right)\right)\right) \times \left(- \sin \left(\cos x\right)\right) \times - \sin \left(x\right)$

= $- \sin \left(\cos \left(\cos \left(x\right)\right)\right) \sin \left(\cos x\right) \sin \left(x\right)$