Question

how would i find the derivative of these?

f(x)=ln((√x)+1)

and

f(x)=ln(x+1/x-1)


thank u!!

Answer 1

first one:
remember that √x = x^(1/2)

f ' (x) = 1/(√x + 1) ( (1/2)x^(-1/2)
= 1/( 2x^(1/2) (√x)(√x + 1)
= 1/( 2√x(√x + 1) or 1/(2(x+√x)

2nd:

not clear if you meant
ln( x + 1/x - 1) --- the way you typed it , or
ln( (x+1)/(x-1) ) --- I have a feeling that's the one, or
ln( x + 1/(x-1) )

I will read it as ln ((x+1)/(x-1))
then this can be changed to
ln(x+1) - ln(x-1)
f ' (x) = 1/(x+1) - 1/(x-1)
= (x-1 - (x+1))/((x+1)(x-1))
= -2/((x+1)(x-1))

notice how important brackets are ?