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For a number in form af(x) while differentiating with dont interfere with the constant a.

So, it is a d(f(x))/dx

Here assuming were differentiating with respect to x,

we have 2*d(tan x)/dx

From here you have options weather you can differentiate ie based on uv or u/v method. I'll be using u/v (easy to apply for division),

Here's the general form,

(u/v)' = {vu' - uv'}/v^2

d(tan x)/dx =d(sinx/cosx )/dx = {cosx(sinx)' - sinx(cosx)'}/(cos^2)

= {(cos^2 x) - (-sin^2x)}/cos^2 = {(cos^2 x) +(sin^2x)}/cos^2

Since sin^2 x + cos^2 x =1 ,

we have

d(tan x)/dx = 1/cos^2x = sec ^2 x.

Therefore the final answer for,

2*d(tan x)/dx = 2*sec^2 x

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Q: How do you differentiate 2tanx?

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tan(2x) = 2tanx/(1 - tan2x) So tan(2x) +1 = 2tanx/(1 - tan2x) + 1 = 2tanx/(1 - tan2x) + (1 - tan2x)/(1 - tan2x) = (-tan2x + 2tanx + 1)/(1 - tan2x)

I assume you mean (tanx+1)^2 In which case, (tanx+1)^2=tan2x+2tanx+1

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