Given f(x)=tanx. Then f′(x) is equal to
sec2x
f(x)=tanx
We can write
tanx as sinxcosx
∴f(x)=sinxcosx=g(x)h(x)
We know,
f′(x)=h(x)g′(x)−h′(x)g(x)(h(x))2
=cosxsin′(x)−cos′(x)sinx(cosx)2
=−cosxcosx−(−sinx)(sinx)(cosx)2
=cos2x+sin2xcos2x
=sec2
∴f′(x)=sec2x ∀ x ∈(−π2,π2)