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## Homework Statement

show that the following functions are differentiable everywhere and then also find f'(z) and f''(z).

(a) f(z) = iz + 2

so f(z) = ix -y +2

then u(x,y) = 2-y, v(x,y) = x

## Homework Equations

z=x+iy

z=u(x,y) +iv(x,y)

Cauchy-Riemann conditions says is differentiable everywhere if :

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

## The Attempt at a Solution

so using the Cauchy-Riemann conditions i find that the function is differentiable everywhere. the part I am stuck on is finding the first derivative.

f'(z) should be in the form of two partial derivatives right? because of the way the variables are set up.

so...

f'(z) =

∂z/∂x = ∂z/∂u(∂u/∂x) + ∂z/∂v(∂v/∂x)

∂z/∂y = ∂z/∂u/(∂u/∂y) + ∂z/∂v(∂v/∂y)

but where do i go from here? i can solve partials of u with respect to x or y but i don't know how to solve the partials of z with respect to u.

thank you!