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y=xe^ three +sinx/lnx
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Similar expressions
y=xe^3-sinx/lnx

The derivative of the function/ y=xe^3+sinx/lnx

Derivative of y=xe^3+sinx/lnx

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

Differentiate $e^{3} x + \frac{\sin{\left(x \right)}}{\log{\left(x \right)}}$ 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: $x$ goes to $1$
  So, the result is: $e^{3}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \log{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  Now plug in to the quotient rule:
  
  $\frac{\log{\left(x \right)} \cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{x}}{\log{\left(x \right)}^{2}}$
The result is: $\frac{\log{\left(x \right)} \cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{x}}{\log{\left(x \right)}^{2}} + e^{3}$
Now simplify:

$e^{3} + \frac{\cos{\left(x \right)}}{\log{\left(x \right)}} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}$

The answer is:

$e^{3} + \frac{\cos{\left(x \right)}}{\log{\left(x \right)}} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 3   cos(x)     sin(x) 
E  + ------ - ---------
     log(x)        2   
              x*log (x)

$$e^{3} + \frac{\cos{\left(x \right)}}{\log{\left(x \right)}} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}$$

Simplify

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)

$$\frac{- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x \log{\left(x \right)}} + \frac{\sin{\left(x \right)}}{x^{2} \log{\left(x \right)}} + \frac{2 \sin{\left(x \right)}}{x^{2} \log{\left(x \right)}^{2}}}{\log{\left(x \right)}}$$

Simplify

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)

$$\frac{- \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x \log{\left(x \right)}} + \frac{3 \cos{\left(x \right)}}{x^{2} \log{\left(x \right)}} + \frac{6 \cos{\left(x \right)}}{x^{2} \log{\left(x \right)}^{2}} - \frac{2 \sin{\left(x \right)}}{x^{3} \log{\left(x \right)}} - \frac{6 \sin{\left(x \right)}}{x^{3} \log{\left(x \right)}^{2}} - \frac{6 \sin{\left(x \right)}}{x^{3} \log{\left(x \right)}^{3}}}{\log{\left(x \right)}}$$

Simplify

The graph

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Derivative of y=xe^3+sinx/lnx

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The solution