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Derivative of:
Derivative of x*e
Derivative of atan(1/x)
Derivative of coth(x)/tanh(x)
Derivative of 6^x
Identical expressions
y=ln(sinxcosx/(e^x)(two ^x))
y equally ln( sinus of x co sinus of e of x divide by (e to the power of x)(2 to the power of x))
y equally ln( sinus of x co sinus of e of x divide by (e to the power of x)(two to the power of x))
y=ln(sinxcosx/(ex)(2x))
y=lnsinxcosx/ex2x
y=lnsinxcosx/e^x2^x
y=ln(sinxcosx divide by (e^x)(2^x))

The derivative of the function/ y=ln(sinxcosx/(e^x)(2^x))

Derivative of y=ln(sinxcosx/(e^x)(2^x))

The solution

You have entered [src]

   /               x\
   |sin(x)*cos(x)*2 |
log|----------------|
   |        x       |
   \       e        /

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

  /   /               x\\
d |   |sin(x)*cos(x)*2 ||
--|log|----------------||
dx|   |        x       ||
  \   \       e        //

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

Transform

Detail solution

Let $u = \frac{2^{x} \sin{\left(x \right)} \cos{\left(x \right)}}{e^{x}}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2^{x} \sin{\left(x \right)} \cos{\left(x \right)}}{e^{x}}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 2^{x} \sin{\left(x \right)} \cos{\left(x \right)}$ and $g{\left(x \right)} = e^{x}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = 2^{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. $\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}$
    $g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    $h{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} h{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    The result is: $- 2^{x} \sin^{2}{\left(x \right)} + 2^{x} \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 2^{x} \cos^{2}{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $e^{x}$ is itself.
  Now plug in to the quotient rule:
  
  $\left(- 2^{x} e^{x} \sin{\left(x \right)} \cos{\left(x \right)} + \left(- 2^{x} \sin^{2}{\left(x \right)} + 2^{x} \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 2^{x} \cos^{2}{\left(x \right)}\right) e^{x}\right) e^{- 2 x}$
The result of the chain rule is:

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

$- 2 \tan{\left(x \right)} - 1 + \log{\left(2 \right)} + \frac{2}{\sin{\left(2 x \right)}}$

The answer is:

$- 2 \tan{\left(x \right)} - 1 + \log{\left(2 \right)} + \frac{2}{\sin{\left(2 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 -x / x    2     -x    x    2     -x    x         -x           x         -x              \  x
2  *\2 *cos (x)*e   - 2 *sin (x)*e   - 2 *cos(x)*e  *sin(x) + 2 *cos(x)*e  *log(2)*sin(x)/*e 
---------------------------------------------------------------------------------------------
                                        cos(x)*sin(x)

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

Simplify

The second derivative [src]

   2         2                                                2         2                                                2         2                                                    2           2           2                  2                                  2                                             /   2         2                                          \       
sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)   sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)   sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)   - 2*sin (x) + 2*cos (x) - 2*cos (x)*log(2) + 2*sin (x)*log(2) + 3*cos(x)*sin(x) - log (2)*cos(x)*sin(x) + 2*cos(x)*log(2)*sin(x)   \sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2)
-------------------------------------------------------- - -------------------------------------------------------- - -------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------
                           2                                                          2                                                    cos(x)*sin(x)                                                                                  cos(x)*sin(x)                                                                                       cos(x)*sin(x)                          
                        sin (x)                                                    cos (x)

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

Simplify

The third derivative [src]

    /   2         2                                          \     /       2           2           2                  2                                  2                                          \     /   2         2                                          \     /       2           2           2                  2                                  2                                          \      2         2           2                  2       2           2       2           2                                   3                                                  2                       2         2                                               /       2           2           2                  2                                  2                                          \     /   2         2                                          \            /   2         2                                          \            /   2         2                                          \            /   2         2                                          \             2    /   2         2                                          \     /   2         2                                          \            /       2           2           2                  2                                  2                                          \       
  2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/   2*\- 2*sin (x) + 2*cos (x) - 2*cos (x)*log(2) + 2*sin (x)*log(2) + 3*cos(x)*sin(x) - log (2)*cos(x)*sin(x) + 2*cos(x)*log(2)*sin(x)/   2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/   2*\- 2*sin (x) + 2*cos (x) - 2*cos (x)*log(2) + 2*sin (x)*log(2) + 3*cos(x)*sin(x) - log (2)*cos(x)*sin(x) + 2*cos(x)*log(2)*sin(x)/   sin (x) - cos (x) - 6*cos (x)*log(2) - 3*log (2)*sin (x) + 3*cos (x)*log (2) + 6*sin (x)*log(2) + 11*cos(x)*sin(x) + log (2)*cos(x)*sin(x) - 9*cos(x)*log(2)*sin(x) - 3*log (2)*cos(x)*sin(x)   sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)   2*\- 2*sin (x) + 2*cos (x) - 2*cos (x)*log(2) + 2*sin (x)*log(2) + 3*cos(x)*sin(x) - log (2)*cos(x)*sin(x) + 2*cos(x)*log(2)*sin(x)/   2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*sin(x)   2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*cos(x)   2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2)   2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2)   log (2)*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/   2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2)   2*\- 2*sin (x) + 2*cos (x) - 2*cos (x)*log(2) + 2*sin (x)*log(2) + 3*cos(x)*sin(x) - log (2)*cos(x)*sin(x) + 2*cos(x)*log(2)*sin(x)/*log(2)
- ------------------------------------------------------------ - ------------------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------ - ------------------------------------------------------------------- - ------------------------------------------------------------------- - ------------------------------------------------------------------- + ------------------------------------------------------------------- - ------------------------------------------------------------------ + ------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------
                               2                                                                                                  2                                                                                                  2                                                                                                  2                                                                                                                                                             cos(x)*sin(x)                                                                                                                cos(x)*sin(x)                                                                                    cos(x)*sin(x)                                                                                                3                                                                     3                                                                     2                                                                     2                                                              cos(x)*sin(x)                                                         cos(x)*sin(x)                                                                                             cos(x)*sin(x)                                                               
                            cos (x)                                                                                            cos (x)                                                                                            sin (x)                                                                                            sin (x)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  cos (x)                                                               sin (x)                                                               sin (x)                                                               cos (x)

$$- \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} - \frac{\left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}^{2}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \frac{2 \left(- 2 \sin^{2}{\left(x \right)} + 2 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \frac{2 \left(- 2 \sin^{2}{\left(x \right)} + 2 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(- 2 \sin^{2}{\left(x \right)} + 2 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(- 2 \sin^{2}{\left(x \right)} + 2 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)}} + \frac{- 3 \log{\left(2 \right)}^{2} \sin^{2}{\left(x \right)} + \sin^{2}{\left(x \right)} + 6 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - 9 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} - 3 \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)}^{3} \sin{\left(x \right)} \cos{\left(x \right)} + 11 \sin{\left(x \right)} \cos{\left(x \right)} - 6 \log{\left(2 \right)} \cos^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + 3 \log{\left(2 \right)}^{2} \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}}$$

Simplify

The graph

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How to use it?

Identical expressions

Derivative of y=ln(sinxcosx/(e^x)(2^x))

The graph:

Piecewise:

The solution