How Can You Simplify the Derivative of Ln(x + 1/x)?

[jorgen]

Hi all,

I have to differentiate

$$Ln(x+\frac{1}{x})$$

where I first differentiate Ln and than multiply by the differentiation of the inner function

$$1/(x+\frac{1}{x})*(1-\frac{1}{x^2}$$

which I simplify to

$$\frac{x}{x^2+1}*(1-\frac{1}{x^2})$$

$$\frac{x}{x^2+1}-\frac{1}{x*(x^2+1)}$$

the problem is I cannot rewrite it to this

$$\frac{2*x}{x^2+1}-\frac{1}{x}$$

how to rewrite it - any help or advise appreciated. Thanks in advance

Best
Jorgen

 

[borgwal]

Start at the final expression, and write it as a fraction with one denominator: x(x^2+1)

 

[jorgen]

thanks,

so I put into one fraction

$$\frac{x^2-1}{x*(x^2+1)}$$

but I don't know how to start rearranging this... Any new hints

Best

J

 

[gabbagabbahey]

$x^2-1=(x^2+1)-2$ ;0)

 

[jorgen]

so I rewrite the fraction using this hint

$$\frac{(x^2+1)-2}{x*(x^2+1)}$$

I split the fraction into

$$\frac{1}{x}-\frac{2}{x*(x^2+1)}$$

but I can still not see how to rearrange it

Thanks in advance

Best
J

 

[gabbagabbahey]

Use partial fractions to decompose your second term; in other words, find the constants $A$,$B$, and $C$ that satisfy:

$$\frac{2}{x*(x^2+1)}=\frac{A}{x}+\frac{Bx+C}{x^2+1}$$

 

[Tedjn]

In what form do you want it? Whether it is condensed into one fraction or written as a difference doesn't matter if both expressions are equal.