# Thread: Help finding derivative using definition (which results in a complex fraction)

1. ## Help finding derivative using definition (which results in a complex fraction)

Hello! This is the last problem of the night (I promise!). Will someone please help me with the following problem; if I could use the quotient rule I wouldn't have had any problems, but here is the question:

Find the derivative of

using the definition of derivative.
State the domain of the function and the domain of its derivative.

When I worked it out, this was what I got:

But the answer I got using Mathway.com was:

I could DEFINITELY use some help. Where did I go wrong?

2. ## Re: Help finding derivative using definition (which results in a complex fraction)

so

$\displaystyle lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \frac{\frac{3+x+h}{1-3x-3h} - \frac{3+x}{1-3x}}{h} = \frac{\frac{(3+x+h)(1-3x)-(3+x)(1-3x-3h)}{(1-3x-3h)(1-3x)}}{h}= \frac{\frac{(3-3)+(x-x)+(9x-9x)+(3x^2-3x^2)+(9h+h)+(3xh-3xh)}{(1-3x-3h)(1-3x)}}{h}= \frac{10h}{h*(1-3x-3h)(1-3x)} = \frac{10}{(1-3x-3h)(1-3x)} = \frac{10}{(1-3x)^2}$

3. ## Re: Help finding derivative using definition (which results in a complex fraction)

I can't believe I made so many basic mistakes. Thank you so much for your help!! :-)