# Thread: Derivative of a complex function using limit definition

1. ## Derivative of a complex function using limit definition

Find the derivative of $\displaystyle f(z)=5z^2-10z+8$ using the following definition:

$\displaystyle f'(z_0)=\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}$

I started with

$\displaystyle f'(z_0)=\lim_{z\to z_0}\frac{5z^2-10z+8-(5z_0^2-10z_0+8)}{z-z_0}$

which simplifies a bit to

$\displaystyle f'(z_0)=\lim_{z\to z_0}\frac{5z^2-10z-5z_0^2+10z_0}{z-z_0}$

but where do you go from there? Thanks!

2. ## Re: Derivative of a complex function using limit definition

Originally Posted by Ragnarok
Find the derivative of $\displaystyle f(z)=5z^2-10z+8$ using the following definition:

$\displaystyle f'(z_0)=\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}$

I started with

$\displaystyle f'(z_0)=\lim_{z\to z_0}\frac{5z^2-10z+8-(5z_0^2-10z_0+8)}{z-z_0}$

which simplifies a bit to

$\displaystyle f'(z_0)=\lim_{z\to z_0}\frac{5z^2-10z-5z_0^2+10z_0}{z-z_0}$

but where do you go from there? Thanks!
$\displaystyle \frac{5z^2-10z-5z_0^2+10z_0}{z-z_0}=\frac{5(z^2-z_0^2)-10(z-z_0)}{z-z_0}$

3. ## Re: Derivative of a complex function using limit definition

Try this:

$\displaystyle f'(z_0)=\lim_{z\to z_0}\frac{5z^2-5z_0^2-10z+10z_0}{z-z_0}$

$\displaystyle f'(z_0)=\lim_{z\to z_0}\frac{5(z+z_0)(z-z_0)-10(z-z_0)}{z-z_0}$

- Hollywood

Awww - Plato, you beat me!

4. ## Re: Derivative of a complex function using limit definition

Ah thank you! I forgot my difference of squares factorization.