Question

How do I simplify this?

Answer 1

\[\frac{ 1+\sqrt{x}-\frac{ 1 }{ 2 }x (x) ^{-1/2} }{ (1+\sqrt{x})^2 }\]

Answer 2

Those are square roots over those 'x's by the way. The answer is supposed to be \[\frac{ 2+ \sqrt{x} }{ 2 (1+\sqrt{x})^2 }\] but I don't understand how they got there.

Answer 3

" Those are square roots over those 'x's " ? can you then please show what the problem really is like?

Answer 4

It was to find the derivative of \[\frac{ x }{ 1+\sqrt{x} }\]

Answer 5

oh

Answer 6

\(\large\color{black}{ \displaystyle \frac{x}{1+\sqrt{x}} }\)

the first option is to use the quotient rule

Answer 7

well, or you can say, \(\large\color{black}{ \displaystyle x(1+\sqrt{x})^{-1} }\) and use the product rule

Answer 8

Yeah I used the quotient rule to get that

Answer 9

or you can do logarithmic differentiation

\(\large\color{black}{ \displaystyle y=\frac{x}{1+\sqrt{x}} }\)

\(\large\color{black}{ \displaystyle \ln (y)=\ln\left( \frac{x}{1+\sqrt{x}}\right) }\)

\(\large\color{black}{ \displaystyle \ln y=\ln x -\ln(1+\sqrt{x}) }\)
and on....

Answer 10

But I was just wondering if you knew how I would simplify something like that?

Answer 11

using the quotient rule

Answer 12

\(\large\color{black}{ \displaystyle \frac{ (x)'(1+\sqrt{x})-(x)(1+\sqrt{x})' }{(1+\sqrt{x})^2} }\)

Answer 13

my latex was not working sorry for the delay

Answer 14

\(\large\color{black}{ \displaystyle \frac{ (x)'(1+\sqrt{x})-(x)(1+\sqrt{x})' }{(1+\sqrt{x})^2} }\)

\(\large\color{black}{ \displaystyle \frac{ (1+\sqrt{x})-(x)(\frac{1}{2\sqrt{x}}) }{(1+\sqrt{x})^2} }\)

Answer 15

\(\large\color{black}{ \displaystyle \frac{ 1+\sqrt{x}-\frac{\sqrt{x}}{2} }{(1+\sqrt{x})^2} }\)

Answer 16

\(\large\color{black}{ \displaystyle \frac{ 1+\frac{2\sqrt{x}}{2}-\frac{\sqrt{x}}{2} }{(1+\sqrt{x})^2} }\)


\(\large\color{black}{ \displaystyle \frac{ 1+\frac{\sqrt{x}}{2} }{(1+\sqrt{x})^2} }\)

Answer 17

or, also

\(\large\color{black}{ \displaystyle \frac{ 1+\frac{\sqrt{x}}{2} }{(1+\sqrt{x})^2} }\)


\(\large\color{black}{ \displaystyle \frac{ \frac{2}{2}+\frac{\sqrt{x}}{2} }{(1+\sqrt{x})^2} }\)


\(\large\color{black}{ \displaystyle \frac{ \frac{2+\sqrt{x}}{2} }{(1+\sqrt{x})^2} }\)


\(\large\color{black}{ \displaystyle \frac{ 2+\sqrt{x}}{2(1+\sqrt{x})^2} }\)

getting the exact same thing as the wolframalpha
http://www.wolframalpha.com/input/?i=derivative+of+x%2F%281%2B%E2%88%9Ax%29

Answer 18

do you understand what I did all throughout the thread ?

Answer 19

Yes!! I finally got it!! Thank you! I figured out the part I kept getting wrong :)

Answer 20

or you showed it to me lol

Answer 21

:)

Answer 22

I did a prove on the quotient using logarithmic differentiation, if you really want.