2 log (x) - 2*x ------------------ 2 log (x) - 2*log(x)
(log(x)^2 - 2*x)/(log(x)^2 - 2*log(x))
Apply the quotient rule, which is:
and .
To find :
Differentiate term by term:
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
The derivative of is .
The result of the chain rule is:
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:
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
The derivative of is .
The result of the chain rule is:
The derivative of a constant times a function is the constant times the derivative of the function.
The derivative of is .
So, the result is:
The result is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
2*log(x) / 2 \ /2 2*log(x)\ -2 + -------- \log (x) - 2*x/*|- - --------| x \x x / ------------------ + ------------------------------ 2 2 log (x) - 2*log(x) / 2 \ \log (x) - 2*log(x)/
/ / 2 \ \ | | 4*(-1 + log(x)) | / 2 \ | | |1 + ---------------------|*\- log (x) + 2*x/ / log(x)\ | | | 2 | 4*|1 - ------|*(-1 + log(x))| | -1 + log(x) \ (-2 + log(x)) *log(x)/ \ x / | 2*|- ----------- - --------------------------------------------- + ----------------------------| \ x x*log(x) (-2 + log(x))*log(x) / ------------------------------------------------------------------------------------------------ x*(-2 + log(x))*log(x)
/ / 2 \ / 3 \\ | / log(x)\ | 4*(-1 + log(x)) | / 2 \ | 12*(-1 + log(x)) 24*(-1 + log(x)) || | 6*|1 - ------|*|1 + ---------------------| \- log (x) + 2*x/*|-5 + 2*log(x) + ---------------- + ----------------------|| | \ x / | 2 | 2 | log(x) 2 2 || |-3 + 2*log(x) \ (-2 + log(x)) *log(x)/ 6*(-1 + log(x)) \ (-2 + log(x)) *log (x)/| 2*|------------- - ------------------------------------------ + ---------------------- + -----------------------------------------------------------------------------| \ x log(x) x*(-2 + log(x))*log(x) x*(-2 + log(x))*log(x) / ----------------------------------------------------------------------------------------------------------------------------------------------------------------------- 2 x *(-2 + log(x))*log(x)