Question

ection (A) : First principle, Basic theorem 1. Find the derivative of following functions with respect to \( x \) by the first principle (ab - initio method). (i) \( f(x)=\sin x^{2} \) (ii) \( f(x)=e^{2 x+3} \)

Answer 1

Solution(i) 

d(sinx²) /dt.     (by using chain rule) differentiation of (sinx)²  = 2. sinx then differentiation of sinx=cosx and differentiation of x = 1(e

=2sinx.(cosx).(1)

= sin2x

Solution (ii) 

d(e^2x+3 ) /dt = e^2x+3   (2+0). {differentiation of 2x+3 by chain rule}

=2.e^2x+3

H.