Answer
$$\frac{{dy}}{{dx}} = 8{\tan ^7}x{\sec ^2}x$$
Work Step by Step
$$\eqalign{
& y = {\tan ^8}x \cr
& {\text{we can write the function as}} \cr
& y = {\left( {\tan x} \right)^4} \cr
& {\text{differentiate with respect to }}x \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {{{\left( {\tan x} \right)}^4}} \right] \cr
& {\text{use the general power rule for derivatives }}\frac{d}{{dx}}\left[ {{u^n}} \right] = n{u^{n - 1}}\frac{{du}}{{dx}}.{\text{ consider }}u = \cos x \cr
& then \cr
& \frac{{dy}}{{dx}} = 8{\left( {\tan x} \right)^{8 - 1}}\frac{d}{{dx}}\left[ {\tan x} \right] \cr
& {\text{use }}{D_x}\left( {\tan x} \right) = {\sec ^2}x \cr
& \frac{{dy}}{{dx}} = 8{\left( {\tan x} \right)^7}\left( {{{\sec }^2}x} \right) \cr
& {\text{simplifying}} \cr
& \frac{{dy}}{{dx}} = 8{\tan ^7}x{\sec ^2}x \cr} $$
