Detail solution
-
Apply the quotient rule, which is:
and .
To find :
-
The derivative of sine is cosine:
To find :
-
The derivative of is .
Now plug in to the quotient rule:
-
Now simplify:
The answer is:
The first derivative
[src]
cos(x) sin(x)
------ - ---------
log(x) 2
x*log (x)
$$\frac{\cos{\left(x \right)}}{\log{\left(x \right)}} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}$$
The second derivative
[src]
/ 2 \
|1 + ------|*sin(x)
2*cos(x) \ log(x)/
-sin(x) - -------- + -------------------
x*log(x) 2
x *log(x)
----------------------------------------
log(x)
$$\frac{- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x \log{\left(x \right)}} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \sin{\left(x \right)}}{x^{2} \log{\left(x \right)}}}{\log{\left(x \right)}}$$
The third derivative
[src]
/ 3 3 \
2*|1 + ------ + -------|*sin(x) / 2 \
| log(x) 2 | 3*|1 + ------|*cos(x)
3*sin(x) \ log (x)/ \ log(x)/
-cos(x) + -------- - ------------------------------- + ---------------------
x*log(x) 3 2
x *log(x) x *log(x)
----------------------------------------------------------------------------
log(x)
$$\frac{- \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x \log{\left(x \right)}} + \frac{3 \cdot \left(1 + \frac{2}{\log{\left(x \right)}}\right) \cos{\left(x \right)}}{x^{2} \log{\left(x \right)}} - \frac{2 \cdot \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right) \sin{\left(x \right)}}{x^{3} \log{\left(x \right)}}}{\log{\left(x \right)}}$$