# Thread: definition of derivative question

1. ## definition of derivative question

Hey getting stuck on this one

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

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

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

what do i do now?

thanks

Hey getting stuck on this one

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

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

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

what do i do now?

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

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

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

$

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

$

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

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

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

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

$

\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
$

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

$

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

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

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