___ \/ x ------ x e + 1
/ ___ \ d |\/ x | --|------| dx| x | \e + 1/
Apply the quotient rule, which is:
and .
To find :
Apply the power rule: goes to
To find :
Differentiate term by term:
The derivative of the constant is zero.
The derivative of is itself.
The result is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
___ x 1 \/ x *e ---------------- - --------- ___ / x \ 2 2*\/ x *\e + 1/ / x \ \e + 1/
/ / x \ \ | ___ | 2*e | x| | \/ x *|1 - ------|*e | | x | x| | | 1 e \ 1 + e / | -|------ + -------------- + ---------------------| | 3/2 ___ / x\ x | \4*x \/ x *\1 + e / 1 + e / --------------------------------------------------- x 1 + e
/ x 2*x \ ___ | 6*e 6*e | x / x \ \/ x *|1 - ------ + ---------|*e | 2*e | x | x 2| 3*|1 - ------|*e x | 1 + e / x\ | | x| 3 3*e \ \1 + e / / \ 1 + e / ------ + --------------- - --------------------------------- - ----------------- 5/2 3/2 / x\ x ___ / x\ 8*x 4*x *\1 + e / 1 + e 2*\/ x *\1 + e / -------------------------------------------------------------------------------- x 1 + e