# Thread: definition of derivative question

1. ## definition of derivative question

Hey getting stuck on this one

Use definition of the derivative to determine $\displaystyle f'(x)$ where $\displaystyle f(x)= \sqrt{3x}$

$\displaystyle \lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h}$

$\displaystyle \lim_{h\rightarrow 0} \frac{\sqrt{3x+3h}-\sqrt{3h}}{h}$

what do i do now?

thanks

Hey getting stuck on this one

Use definition of the derivative to determine $\displaystyle f'(x)$ where $\displaystyle f(x)= \sqrt{3x}$

$\displaystyle \lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h}$

$\displaystyle \lim_{h\rightarrow 0} \frac{\sqrt{3x+3h}-{\color{red} \sqrt{3h}}}{h}$

what do i do now?

thanks
the red should be $\displaystyle \sqrt{3x}$

to find the limit,start by multiplying by $\displaystyle \frac{\sqrt{3x+3h}+\sqrt{3x}}{\sqrt{3x+3h}+\sqrt{3 x}}$

3. So I got it out to this.
Is this correct?
Thanks

$\displaystyle \lim_{h\rightarrow 0} \frac{\sqrt{3x+3h}-\sqrt{3h}}{h}* \frac{\sqrt{3x+3h}+\sqrt{3x}}{\sqrt{3x+3h}+\sqrt{3 x}}$

$\displaystyle \lim_{h\rightarrow 0} 3*\frac{1}{\sqrt{3x+3h}+\sqrt{3x}}$

$\displaystyle (h\neq 0) \frac{3}{\sqrt{3x+0}+\sqrt{3x}}$

$\displaystyle = \frac{3}{2\sqrt{3x}}$

So I got it out to this.
Is this correct?
Thanks

$\displaystyle \lim_{h\rightarrow 0} \frac{\sqrt{3x+3h}-\sqrt{3h}}{h}* \frac{\sqrt{3x+3h}+\sqrt{3x}}{\sqrt{3x+3h}+\sqrt{3 x}}$
Typographical error- this should be
$\displaystyle \lim_{h\rightarrow 0} \frac{\sqrt{3x+3h}-\sqrt{3x}}{h}* \frac{\sqrt{3x+3h}+\sqrt{3x}}{\sqrt{3x+3h}+\sqrt{3 x}}$

$\displaystyle \lim_{h\rightarrow 0} 3*\frac{1}{\sqrt{3x+3h}+\sqrt{3x}}$

$\displaystyle (h\neq 0) \frac{3}{\sqrt{3x+0}+\sqrt{3x}}$

$\displaystyle = \frac{3}{2\sqrt{3x}}$
Yes, that is exactly right!