_________ / 2*x + 3 / ------- \/ 2*x - 3
sqrt((2*x + 3)/(2*x - 3))
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Apply the quotient rule, which is:
and .
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:
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:
The result of the chain rule is:
Now simplify:
The answer is:
_________ / 2*x + 3 / 1 2*x + 3 \ / ------- *|------- - ----------|*(2*x - 3) \/ 2*x - 3 |2*x - 3 2| \ (2*x - 3) / ---------------------------------------------- 2*x + 3
/ 3 + 2*x \ __________ | 1 - --------| / 3 + 2*x / 3 + 2*x \ | 2 2 -3 + 2*x| / -------- *|1 - --------|*|- -------- - ------- + ------------| \/ -3 + 2*x \ -3 + 2*x/ \ -3 + 2*x 3 + 2*x 3 + 2*x / ------------------------------------------------------------------- 3 + 2*x
/ 2 \ | / 3 + 2*x \ / 3 + 2*x \ / 3 + 2*x \ | __________ | |1 - --------| 6*|1 - --------| 6*|1 - --------| | / 3 + 2*x / 3 + 2*x \ | 8 8 \ -3 + 2*x/ \ -3 + 2*x/ 8 \ -3 + 2*x/ | / -------- *|1 - --------|*|----------- + ---------- + --------------- - ---------------- + -------------------- - --------------------| \/ -3 + 2*x \ -3 + 2*x/ | 2 2 2 2 (-3 + 2*x)*(3 + 2*x) (-3 + 2*x)*(3 + 2*x)| \(-3 + 2*x) (3 + 2*x) (3 + 2*x) (3 + 2*x) / ------------------------------------------------------------------------------------------------------------------------------------------- 3 + 2*x