/ ________\ | / 2 | log\x + \/ x + 1 /
/ / ________\\ d | | / 2 || --\log\x + \/ x + 1 // dx
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Differentiate term by term:
Apply the power rule: goes to
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Differentiate term by term:
Apply the power rule: goes to
The derivative of the constant is zero.
The result is:
The result of the chain rule is:
The result is:
The result of the chain rule is:
Now simplify:
The answer is:
x 1 + ----------- ________ / 2 \/ x + 1 --------------- ________ / 2 x + \/ x + 1
/ 2\ | 2 / x \ | | x |1 + -----------| | |-1 + ------ | ________| | | 2 | / 2 | | | 1 + x \ \/ 1 + x / | -|----------- + ------------------| | ________ ________ | | / 2 / 2 | \\/ 1 + x x + \/ 1 + x / ------------------------------------ ________ / 2 x + \/ 1 + x
3 / 2 \ / x \ / 2 \ / x \ | x | 2*|1 + -----------| | x | 3*|1 + -----------|*|-1 + ------| | ________| 3*x*|-1 + ------| | ________| | 2| | / 2 | | 2| | / 2 | \ 1 + x / \ \/ 1 + x / \ 1 + x / \ \/ 1 + x / -------------------- + ----------------- + --------------------------------- 2 3/2 ________ / ________\ / ________\ / 2\ / 2 | / 2 | | / 2 | \1 + x / \/ 1 + x *\x + \/ 1 + x / \x + \/ 1 + x / ---------------------------------------------------------------------------- ________ / 2 x + \/ 1 + x