Explain why d/dx tan^2 (x) is equal to 2(tan x). d/dx(tan x)

there's a differentiating technique in calculus called the chain rule

note that tan^2(x) = (tan(x))^2 it's just another way of writing it.

y = (tan(x))^2
put u = tan(x)
that means y = u^2

dy/dx = dy/du * du/dx
and if you want to know why this works it's because the "du" in each fraction will cancel with the other leaving us with
dy/dx = dy/dx

power and chain rule

the derivative of x^3 is equal to (3X^2)1

the 1 comes from the derivative of x the same thing happens with tan^2(x)

the 2 comes down to become a coefficient

2tan(x)

then the chain rule says you need to multiply by the derivative of tan(x)

which means the derivative is

2tan(x)sec^2(x)

Chain rule with u = tan^2 x