can you find the derivative of y = ln|1 + x| ? Is there a caught when we have somehing like |....| ?

Question

can you find the derivative of y = ln|1 + x| ?  Is there a caught when we have somehing like |....| ?

anonymous · Answer

$$\Large\frac{d }{dx} \ln f(x)=\frac{ f'(x) }{ f(x) }$$

anonymous · Answer

there  |...| is unneccessary since the defination of ln functi0on desires it to be positive

anonymous · Answer

(lnabs(1+x))=(ln(x+1) )'=1/(x+1)

anonymous · Answer

What is the anti-derivative of 1/x?

anonymous · Answer

lnabsx

anonymous · Answer

exactly

anonymous · Answer

Then if I differentiate that, I get back to 1/x.

Similarly, what would be the antiderivative of 1/x+1?
What would be the derivative of this antiderivative that you just got?

anonymous · Answer

@Christos

raden · Answer

y = |1+x|
y = sqrt((1+x)^2)
squares to both sides :
y^2 = (1+x)^2
now,
take the derivative by using implicit,
2y dy = 2(1+x)(1)dx
dy/dx = 2(1+x)/2y
dy/dx = (1+x)/y
subtitute back, that y = |1+x|
so,
dy/dx = (1+x)/|1+x|

raden · Answer

does that make sense  ?

christos · Answer

thank you