simple differentiation

[red and white kop!]

I'm supposed to differentiate [(sqrt(x))+1]^2/x
I thought this would be fairly straightforward even though I haven't learnt how to differentiate multiplications and divisions of functions.
So I developed [(sqrt(x))+1]^2 to end up with x+2(sqrt(x))+1, which gives me the three terms x/x + 2(sqrt(x))/x + 1/x, which differentiating separately gives me the final result:
-1/x(sqrt(x)) - 1/(x^2)
But this is incorrect apparently, where did I go wrong?

 

[galactus]

Is this what you mean:

\(\displaystyle \displaystyle \frac{(\sqrt{x}+1)^{2}}{x}\)?.

If so, try the quotient rule. To differentiate something like \(\displaystyle \frac{f(x)}{g(x)}\), we use

\(\displaystyle \frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^{2}}\).

In this case, we have \(\displaystyle f(x)=(\sqrt{x}+1)^{2}, \;\ g(x)=x\). Proceed from there and we get:

\(\displaystyle \displaystyle \frac{x\left(\frac{\sqrt{x}+1}{\sqrt{x}}\right)-(\sqrt{x}+1)^{2}(1)}{x^{2}}\)

Now, simplify:

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


Or, we could expand the numerator and divide by x to simplify, then differentiate:

\(\displaystyle \frac{x+2\sqrt{x}+1}{x}\)

\(\displaystyle \frac{2}{\sqrt{x}}+\frac{1}{x}+1\)

Differentiate separately:

\(\displaystyle \frac{-1}{x^{3/2}}-\frac{1}{x^{2}}\)

Write over a common denominator:

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

Same as before.

 

[red and white kop!]

the answer given is
1 - 1/(x(sqrt(x))) - 2/(x^3)
Is this just another form for the same result, or a legit mistake in the textbook?

 

[galactus]

No, that's not the same result.

What they give is the derivative for \(\displaystyle \frac{x^{3}+2x^{3/2}+1}{x^{2}}=x+\frac{2}{\sqrt{x}}+\frac{1}{x^{2}}\).


Did I interpret the problem correctly?. I want to make double sure we are correct before saying the book is wrong.


Might I also make a suggestion?. Since you frequent the site, try learning a little LaTex. It will display things clearer. It's very easy and is worth the effort.

 

[Deleted member 4993]

\(\displaystyle \dfrac{(\sqrt{x} + 1)^2}{x}\)

\(\displaystyle = \ \dfrac{1 + 2\sqrt{x} + x}{x}\)

\(\displaystyle = \ \dfrac{1}{x} + 2\dfrac{1}{\sqrt{x}} + 1\)

differentiating:

\(\displaystyle f'(x) \ = \ -\dfrac{1}{x^2} - \dfrac{1}{x\sqrt{x}} + 0\)

\(\displaystyle f'(x) \ = \ -\dfrac{1 + \sqrt{x}}{x^2}\)

 

[red and white kop!]

yes that's the expression, thanks for your help. i'm always telling myself i'll get around to using la-tex....