____________ / log(2*x) / x*-------- \/ log(10)
sqrt(x*(log(2*x)/log(10)))
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Apply the quotient rule, which is:
and .
To find :
Apply the product rule:
; to find :
Apply the power rule: goes to
; to find :
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
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 of the chain rule is:
The result is:
To find :
The derivative of the constant is zero.
Now plug in to the quotient rule:
The result of the chain rule is:
The answer is:
____________ / x*log(2*x) / 1 log(2*x)\ / ---------- *|--------- + ---------|*log(10) \/ log(10) \2*log(10) 2*log(10)/ ------------------------------------------------ x*log(2*x)
/ 2 \ ____________ | (1 + log(2*x)) 2*(1 + log(2*x))| \/ x*log(2*x) *|-2*log(2*x) + --------------- - ----------------| \ log(2*x) log(2*x) / ----------------------------------------------------------------- 2 _________ 4*x *\/ log(10) *log(2*x)
/ 2 2 3 \ ____________ | 1 1 1 + log(2*x) 3*(1 + log(2*x)) 3*(1 + log(2*x)) (1 + log(2*x)) 9*(1 + log(2*x)) | \/ x*log(2*x) *|- - - -------- + ------------ - ----------------- - ----------------- + --------------- + ---------------- + log(2*x)| | 2 log(2*x) 2 4*log(2*x) 2 2 4*log(2*x) | \ log (2*x) 4*log (2*x) 8*log (2*x) / -------------------------------------------------------------------------------------------------------------------------------------- 3 _________ x *\/ log(10) *log(2*x)