Answer these questions please.

Question

Find the differentiation of the function 𝑓(𝑥)=tan𝑥 using the first principal method.

Answer 1

I assume that what you mean by this is to work out the derivative of the tangent starting from the difference quotient

d (tan(x))/dx = lim [tan(x + h) - tan(x) ]/ h { limit as h → 0 }

When this is worked out using the angle addition formula for tangent :

tan( x + h) = [ tan(x) + tan(h) / [ 1 - tan(x) tan(h) ]]

It is found that the result is indeed sec² (x) provided that

lim tan(h) / h = 1. { limit as h →0 }

This is the case. It follows from lim sin(h) / h = 1

This latter limit cannot be proven by purely algebraic methods. The proof is from geometry. It involves the approach of a regular n-gon to circle as n → ∞