Calculate the derivative of e^(z^2)

Question : Calculate the derivative of e^(z^2)

Solution:

Given function

h(z) = e^(z^2)

Put w=f(z) =z^2 and g(w)=e^w then

h(z)=g(f(z))

Here  h(z) = g(f(z)) is the composition of functions  f(z) and  g(w). Since f(z)=z^2 is differentiable at any complex number z and g(w)=e^w is differentiable at any complex number w Hence h(z) is also differentiable at at any point z and its derivative is given as

 h’(z) = g’(w)f’(z) = g’(f(z))f’(z)

where derivatives of g and f are

 g’(w)=d/dw(e^w)=e^w and f’(z) =d/dz(z^2)=2z respectively.

Therefore derivative of e^(z^2) is 2ze^(z^2)