/ _________\ | / 2 2 | log\x + \/ x + a /
/ / _________\\ d | | / 2 2 || --\log\x + \/ x + a // 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 2 \/ x + a ---------------- _________ / 2 2 x + \/ x + a
/ 2 \ |/ x \ 2 | ||1 + ------------| x | || _________| -1 + -------| || / 2 2 | 2 2| |\ \/ a + x / a + x | -|------------------- + ------------| | _________ _________| | / 2 2 / 2 2 | \ x + \/ a + x \/ a + x / -------------------------------------- _________ / 2 2 x + \/ a + x
3 / 2 \ / x \ / 2 \ / x \ | x | 2*|1 + ------------| | x | 3*|1 + ------------|*|-1 + -------| | _________| 3*x*|-1 + -------| | _________| | 2 2| | / 2 2 | | 2 2| | / 2 2 | \ a + x / \ \/ a + x / \ a + x / \ \/ a + x / --------------------- + ------------------ + ----------------------------------- 2 3/2 / _________\ _________ / _________\ / 2 2\ | / 2 2 | / 2 2 | / 2 2 | \a + x / \x + \/ a + x /*\/ a + x \x + \/ a + x / -------------------------------------------------------------------------------- _________ / 2 2 x + \/ a + x