log(x) + 1 ---------- 2 x *sin(x)
(log(x) + 1)/((x^2*sin(x)))
Apply the quotient rule, which is:
and .
To find :
Differentiate term by term:
The derivative of the constant is zero.
The derivative of is .
The result is:
To find :
Apply the product rule:
; to find :
Apply the power rule: goes to
; to find :
The derivative of sine is cosine:
The result is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
/ 1 \ |---------| | 2 | / 2 \ \x *sin(x)/ \- x *cos(x) - 2*x*sin(x)/*(log(x) + 1) ----------- + --------------------------------------- x 4 2 x *sin (x)
/ 2 \ |/2 cos(x)\ 2*sin(x) - x *sin(x) + 4*x*cos(x) 2*(2*sin(x) + x*cos(x)) (2*sin(x) + x*cos(x))*cos(x)| (1 + log(x))*||- + ------|*(2*sin(x) + x*cos(x)) - --------------------------------- + ----------------------- + ----------------------------| 1 \\x sin(x)/ x x sin(x) / 2*(2*sin(x) + x*cos(x)) - - + ---------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------- x sin(x) x*sin(x) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ 3 x *sin(x)
/ /2 cos(x)\ / 2 \ /2 cos(x)\ /2 cos(x)\ \ | / 2 \ 2 / 2 \ |- + ------|*\2*sin(x) - x *sin(x) + 4*x*cos(x)/ 2*|- + ------|*(2*sin(x) + x*cos(x)) 2 |- + ------|*(2*sin(x) + x*cos(x))*cos(x) / 2 \ | | | 6 2*cos (x) 4*cos(x)| -6*cos(x) + x *cos(x) + 6*x*sin(x) 6*\2*sin(x) - x *sin(x) + 4*x*cos(x)/ 10*(2*sin(x) + x*cos(x)) \x sin(x)/ \x sin(x)/ 3*cos (x)*(2*sin(x) + x*cos(x)) \x sin(x)/ 3*\2*sin(x) - x *sin(x) + 4*x*cos(x)/*cos(x) 8*(2*sin(x) + x*cos(x))*cos(x)| / 2 \ (1 + log(x))*|2*sin(x) + x*cos(x) + (2*sin(x) + x*cos(x))*|1 + -- + --------- + --------| - ---------------------------------- - ------------------------------------- + ------------------------ - ------------------------------------------------ + ------------------------------------ + ------------------------------- + ----------------------------------------- - -------------------------------------------- + ------------------------------| |/2 cos(x)\ 2*sin(x) - x *sin(x) + 4*x*cos(x) 2*(2*sin(x) + x*cos(x)) (2*sin(x) + x*cos(x))*cos(x)| | | 2 2 x*sin(x)| x 2 2 x x 2 sin(x) x*sin(x) x*sin(x) | 3*||- + ------|*(2*sin(x) + x*cos(x)) - --------------------------------- + ----------------------- + ----------------------------| 2 \ \ x sin (x) / x x sin (x) / \\x sin(x)/ x x sin(x) / 3*(2*sin(x) + x*cos(x)) -- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- + ----------------------- 2 sin(x) x*sin(x) 2 x x *sin(x) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 3 x *sin(x)