d/dx xarcsinh(1/x) ... Why does answer have absolute value?

[UsafaMufasa]

I am self-learning hyperbolic functions and the problem states to find f'(x) if

f(x) = xsinh^-1(1/x)

Here is my work:

f'(x) = sinh^-1(1/x) + (-x)/[(x²)(1+1/x²)^1/2]

I simplified it to:

f'(x) = sinh^-1(1/x) - 1/(x²+1)^1/2

However, the answer the book gives is:

f'(x) = sinh^-1(1/x) - |x|/[x(x²+1)^1/2]

I am not understanding where the absolute value is coming from. Any help is appreciated.

btw... this is my first post so if I did something incorrectly please let me know:shock:

 

[Steven G]

I am self-learning hyperbolic functions and the problem states to find f'(x) if

f(x) = xsinh^-1(1/x)

Here is my work:

f'(x) = sinh^-1(1/x) + (-x)/[(x²)(1+1/x²)^1/2]

I simplified it to:

f'(x) = sinh^-1(1/x) - 1/(x²+1)^1/2

However, the answer the book gives is:

f'(x) = sinh^-1(1/x) - |x|/[x(x²+1)^1/2]

I am not understanding where the absolute value is coming from. Any help is appreciated.

btw... this is my first post so if I did something incorrectly please let me know:shock:

(1+1/x²)^1/2 =((x²+1)/x²)^1/2 =(x²+1)^1/2 /(x²)^1/2
This denominator is |x|, NOT x. After all ((-3)²)^1/2 is not -3, it is |-3|=3!

 

[UsafaMufasa]

Thanks Jomo! The way you rewrote that expression clarifies where the absolute value comes into play.