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y=xe^3+sinx/lnx

Derivative of y=xe^3+sinx/lnx

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   3   sin(x)
x*E  + ------
       log(x)
$$e^{3} x + \frac{\sin{\left(x \right)}}{\log{\left(x \right)}}$$
x*E^3 + sin(x)/log(x)
Detail solution
  1. Differentiate term by term:

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the power rule: goes to

      So, the result is:

    2. Apply the quotient rule, which is:

      and .

      To find :

      1. The derivative of sine is cosine:

      To find :

      1. The derivative of is .

      Now plug in to the quotient rule:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
 3   cos(x)     sin(x) 
E  + ------ - ---------
     log(x)        2   
              x*log (x)
$$e^{3} + \frac{\cos{\left(x \right)}}{\log{\left(x \right)}} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}$$
The second derivative [src]
            sin(x)    2*cos(x)    2*sin(x) 
-sin(x) + --------- - -------- + ----------
           2          x*log(x)    2    2   
          x *log(x)              x *log (x)
-------------------------------------------
                   log(x)                  
$$\frac{- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x \log{\left(x \right)}} + \frac{\sin{\left(x \right)}}{x^{2} \log{\left(x \right)}} + \frac{2 \sin{\left(x \right)}}{x^{2} \log{\left(x \right)}^{2}}}{\log{\left(x \right)}}$$
The third derivative [src]
           6*sin(x)     6*sin(x)     2*sin(x)   3*sin(x)    3*cos(x)    6*cos(x) 
-cos(x) - ---------- - ---------- - --------- + -------- + --------- + ----------
           3    3       3    2       3          x*log(x)    2           2    2   
          x *log (x)   x *log (x)   x *log(x)              x *log(x)   x *log (x)
---------------------------------------------------------------------------------
                                      log(x)                                     
$$\frac{- \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x \log{\left(x \right)}} + \frac{3 \cos{\left(x \right)}}{x^{2} \log{\left(x \right)}} + \frac{6 \cos{\left(x \right)}}{x^{2} \log{\left(x \right)}^{2}} - \frac{2 \sin{\left(x \right)}}{x^{3} \log{\left(x \right)}} - \frac{6 \sin{\left(x \right)}}{x^{3} \log{\left(x \right)}^{2}} - \frac{6 \sin{\left(x \right)}}{x^{3} \log{\left(x \right)}^{3}}}{\log{\left(x \right)}}$$
The graph
Derivative of y=xe^3+sinx/lnx