3 / 2 \ \x - 1/ ----------- 2 / 3 \ \4*x - 5/
(x^2 - 1)^3/(4*x^3 - 5)^2
Apply the quotient rule, which is:
and .
To find :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Differentiate term by term:
The derivative of the constant is zero.
Apply the power rule: goes to
The result is:
The result of the chain rule is:
To find :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
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:
The result of the chain rule is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
3 2 2 / 2 \ / 2 \ 24*x *\x - 1/ 6*x*\x - 1/ - --------------- + ------------- 3 2 / 3 \ / 3 \ \4*x - 5/ \4*x - 5/
/ 2 / 3 \\ | / 2\ | 18*x || | 8*x*\-1 + x / *|-1 + ---------|| | 3 / 2\ | 3|| / 2\ | 2 48*x *\-1 + x / \ -5 + 4*x /| 6*\-1 + x /*|-1 + 5*x - --------------- + -------------------------------| | 3 3 | \ -5 + 4*x -5 + 4*x / --------------------------------------------------------------------------- 2 / 3\ \-5 + 4*x /
/ 3 / 3 6 \ \ | / 2\ | 108*x 864*x | 2 / 3 \| | 2*\-1 + x / *|1 - --------- + ------------| 2 / 2\ | 18*x || | | 3 2| 36*x *\-1 + x / *|-1 + ---------|| | | -5 + 4*x / 3\ | 2 / 2\ / 2\ | 3|| | / 2\ \ \-5 + 4*x / / 18*x *\-1 + x /*\-1 + 5*x / \ -5 + 4*x /| 24*|x*\-3 + 5*x / - ------------------------------------------- - --------------------------- + ---------------------------------| | 3 3 3 | \ -5 + 4*x -5 + 4*x -5 + 4*x / ---------------------------------------------------------------------------------------------------------------------------------- 2 / 3\ \-5 + 4*x /