___ 1 + \/ x --------- ___ 1 - \/ x
/ ___\ d |1 + \/ x | --|---------| dx| ___| \1 - \/ x /
Apply the quotient rule, which is:
and .
To find :
Differentiate term by term:
The derivative of the constant is zero.
Apply the power rule: goes to
The result is:
To find :
Differentiate term by term:
The derivative of the constant is zero.
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
___ 1 1 + \/ x ------------------- + -------------------- ___ / ___\ 2 2*\/ x *\1 - \/ x / ___ / ___\ 2*\/ x *\1 - \/ x /
/ ___\ / 1 2 \ \1 + \/ x /*|---- + --------------| | 3/2 / ___\| 1 2 \x x*\-1 + \/ x // ---- + -------------- - ----------------------------------- 3/2 / ___\ ___ x x*\-1 + \/ x / -1 + \/ x ----------------------------------------------------------- / ___\ 4*\-1 + \/ x /
/ / ___\ / 1 2 2 \ \ | \1 + \/ x /*|---- + --------------- + ------------------| 1 2 | | | 5/2 2 / ___\ 2| ---- + --------------| | |x x *\-1 + \/ x / 3/2 / ___\ | 3/2 / ___\| | 1 1 \ x *\-1 + \/ x / / x x*\-1 + \/ x /| 3*|- ---- - --------------- + --------------------------------------------------------- - ---------------------| | 5/2 2 / ___\ ___ ___ / ___\ | \ x x *\-1 + \/ x / -1 + \/ x \/ x *\-1 + \/ x / / ---------------------------------------------------------------------------------------------------------------- / ___\ 8*\-1 + \/ x /