# Thread: A few Questions on the derivative

1. ## A few Questions on the derivative

Hello, I had a few questions that I completed and I'm unsure if my answers are correct. i had to find the derivative of the √2x, using first principle, f(a+h) - f(a)/h. The answer that I arrived at is 1/√2√x, but when i use the power rule i get 1/x^2. Are they both wrong, or am i doing something wrong, somewhere? The second question is finding the derivative of 3x/(x^2 + 4), the answer i arrived at, is 3/x^2 + 4 - 6x^2/(x^2 + 4)^2. I got this using the quotient rule. Sorry about the mess, i'm completly new to calculus, have no one around me to check my answers until monday and, I just want to know my mistakes. Again, thanks very much.

2. Hi.

Originally Posted by walk_in_fark
Hello, I had a few questions that I completed and I'm unsure if my answers are correct. i had to find the derivative of the √2x, using first principle, f(a+h) - f(a)/h. The answer that I arrived at is 1/√2√x, but when i use the power rule i get 1/x^2. Are they both wrong, or am i doing something wrong, somewhere?
No, the first solution is correct, I guess you mean $\displaystyle f'(x) = \frac{1}{\sqrt 2 \sqrt x}$, don't you?

$\displaystyle f(x) = \sqrt{2x} = \sqrt{2} \sqrt{x} = \sqrt{2} x^{1/2}$

Thus
$\displaystyle f'(x) = \sqrt 2 \frac 1 2 x^{-1/2} =\frac{ \sqrt 2}{2} * \frac{1}{x^{0.5}}$

$\displaystyle =\frac{1}{\sqrt 2 \sqrt{x}}$

Originally Posted by walk_in_fark
The second question is finding the derivative of 3x/(x^2 + 4), the answer i arrived at, is 3/x^2 + 4 - 6x^2/(x^2 + 4)^2. I got this using the quotient rule. Sorry about the mess, i'm completly new to calculus, have no one around me to check my answers until monday and, I just want to know my mistakes. Again, thanks very much.
Yes, that is correct, but I prefer

$\displaystyle f(x) = \frac{3x}{x^2+4}$

$\displaystyle f'(x) = \frac{3(4 - x^2)}{(x^2 + 4)^2}$ (this is the same solution...)

kind regards
Rapha

3. Originally Posted by walk_in_fark
Hello, I had a few questions that I completed and I'm unsure if my answers are correct. i had to find the derivative of the √2x, using first principle, f(a+h) - f(a)/h. The answer that I arrived at is 1/√2√x, but when i use the power rule i get 1/x^2. Are they both wrong, or am i doing something wrong, somewhere? The second question is finding the derivative of 3x/(x^2 + 4), the answer i arrived at, is 3/x^2 + 4 - 6x^2/(x^2 + 4)^2. I got this using the quotient rule. Sorry about the mess, i'm completly new to calculus, have no one around me to check my answers until monday and, I just want to know my mistakes. Again, thanks very much.
first one
I prefer to write it like

$\displaystyle \sqrt{2x}=(2x)^{\frac{1}{2}}$

$\displaystyle \lim_{h\rightarrow0}\frac{f(a+h)-f(a)}{h}$

$\displaystyle \lim_{h\rightarrow0}\frac{\sqrt{x+h}-\sqrt{x}}{h}=\lim_{h\rightarrow0}\left(\frac{\sqrt {x+h}-\sqrt{x}}{h}\right)\left(\frac{\sqrt{x+h}+\sqrt{x} }{\sqrt{x+h}+\sqrt{x}}\right)$

$\displaystyle \lim_{h\rightarrow0}\left(\frac{x+h-x}{h(\sqrt{x+h}+\sqrt{x})}\right)=\lim_{h\rightarr ow0}\left(\frac{h}{h(\sqrt{x+h}+\sqrt{x})}\right)$

the rest for you just sub zero instead of h after delete h from the denominator and the numentor

4. Originally Posted by walk_in_fark
3x/(x^2 + 4), the answer i arrived at, is 3/x^2 + 4 - 6x^2/(x^2 + 4)^2. .
$\displaystyle f(x)=\frac{3x}{x^2+4}=(3x)(x^2+4)^{-1}$

use product rule

$\displaystyle H(x)=g(x)f(x)....H'(x)=g'(x)f(x)+f'(x)g(x)$