---
title: "Jawab Soal Fungsi Kompleks Turunan"
url: https://mathdium.com/posts/jawab-soal-fungsi-kompleks-turunan
category: "Matematika Universitas"
author: "Annisa Toriq"
published: 2024-07-18
description: "Diketahui $f(z) = -3z^2$ Ubahlah ke bentuk $f(z) = u(x,y)+iv(x,y); z=x+yi$ Buktikan $f'(z) = -6z$ dengan $f'(z) = ux(x,y) + ivx(x,y)$"
---

# Jawab Soal Fungsi Kompleks Turunan

Diketahui $f(z) = -3z^2$

1. Ubahlah ke bentuk $f(z) = u(x,y)+iv(x,y) ; z=x+yi$
2. Buktikan $f'(z) = -6z$ dengan  $f'(z) = ux(x,y) + ivx(x,y)$

**Penyelesaian:**

1. Ubahlah ke bentuk $f(z) = u(x,y)+iv(x,y) ; z=x+yi$ $f(z) = -3z^2 = -3 (x+yi)^2 = -3(x^2+2xyi-y^2) $  
$f(z) = -3x^2-6xyi-3y^2$  
Maka:  
$f(z) = u(x,y)+iv(x,y)$  
$f(z) = (-3x^2 +3y^2) + i(-6xy)$  
$f(z) = -3(x^2 -y^2) + i(-6xy)$
2. Buktikan $f'(z) = -6z$ dengan  $f'(z) = ux(x,y) + ivx(x,y)$ $f(z) = u(x,y)+iv(x,y) $  
$f(z) =-3x^2 +3y^2 + i(-6xy)$Maka:  
$u(x,y) = -3x^2 +3y^2; ux(x,y) = -6x$  
$v(x,y) =-6xy; vx(x,y) = -6y$Sehingga didapat:  
$f'(z) = ux(x,y) + ivx(x,y)$  
$f'(z) = (-6x) + i (-6y)$  
$f'(z) = -6 (x+yi)$  
$f'(z) = -6z$Terbukti $f'(z) = -6z$ dengan  $f'(z) = ux(x,y) + ivx(x,y)$