Question

Differentiate _
a) tan^2 x​

Answer 1

${ (tan) }^{2} x$

Function tan2(x) is the composition f(g(x)) of two functions f(u)=u2 and u=g(x)=tan(x).

Apply the chain rule (f(g(x)))′=ddu(f(u))⋅(g(x))′:

(tan2(x))′=ddu(u2)ddx(tan(x))

Apply the power rule ddu(un)=n⋅u−1+n with n=2:

ddu(u2)ddx(tan(x))=(2u−1+2)ddx(tan(x))=2uddx(tan(x))

Return to the old variable:

2uddx(tan(x))=2tan(x)ddx(tan(x))

The derivative of tangent is ddx(tan(x))=sec2(x):

2tan(x)ddx(tan(x))=2tan(x)sec2(x)

Thus, (tan2(x))′=2tan(x)sec2(x)

Answer: (tan2(x))′=2tan(x)sec2(x)