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Derivative of:
Derivative of 3^-x
Derivative of (3*x-5)^6
Derivative of 1-cost
Derivative of (x+2)*(x^2-4*x+5)
Integral of d{x}:
1/(e^x*sinx)
Identical expressions
one /(e^x*sinx)
1 divide by (e to the power of x multiply by sinus of x)
one divide by (e to the power of x multiply by sinus of x)
1/(ex*sinx)
1/ex*sinx
1/(e^xsinx)
1/(exsinx)
1/exsinx
1/e^xsinx
1 divide by (e^x*sinx)

The derivative of the function/ 1/(e^x*sinx)

Derivative of 1/(e^x*sinx)

The solution

You have entered [src]

    1    
---------
 x       
E *sin(x)

$$\frac{1}{e^{x} \sin{\left(x \right)}}$$

1/(E^x*sin(x))

Detail solution

Let $u = e^{x} \sin{\left(x \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} e^{x} \sin{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = e^{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of $e^{x}$ is itself.
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}$
The result of the chain rule is:

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

$- \frac{\sqrt{2} e^{- x} \sin{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x \right)}}$

The answer is:

$- \frac{\sqrt{2} e^{- x} \sin{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  -x                                
 e     /          x    x       \  -x
------*\- cos(x)*e  - e *sin(x)/*e  
sin(x)                              
------------------------------------
               sin(x)

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

Simplify

The second derivative [src]

/          /    cos(x)\                     (cos(x) + sin(x))*cos(x)         \  -x
|-cos(x) + |1 + ------|*(cos(x) + sin(x)) + ------------------------ + sin(x)|*e  
\          \    sin(x)/                              sin(x)                  /    
----------------------------------------------------------------------------------
                                        2                                         
                                     sin (x)

$$\frac{\left(\left(1 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{- x}}{\sin^{2}{\left(x \right)}}$$

Simplify

The third derivative [src]

/                                                                                                                                                                 /       2            \                                                                \    
|                                                                                                                                                                 |    cos (x)   cos(x)|                                                                |    
|                                                 /    cos(x)\                                                                                2*(cos(x) + sin(x))*|1 + ------- + ------|     /    cos(x)\          /    cos(x)\                         |    
|                                          2      |1 + ------|*(cos(x) + sin(x))                                     2                                            |       2      sin(x)|   2*|1 + ------|*cos(x)   |1 + ------|*(cos(x) + sin(x))*cos(x)|    
|    3*(cos(x) + sin(x))   4*cos(x)   6*cos (x)   \    sin(x)/                     4*(cos(x) + sin(x))*cos(x)   3*cos (x)*(cos(x) + sin(x))                       \    sin (x)         /     \    sin(x)/          \    sin(x)/                         |  -x
|2 - ------------------- + -------- + --------- - ------------------------------ - -------------------------- - --------------------------- - ------------------------------------------ + --------------------- - -------------------------------------|*e  
|           sin(x)          sin(x)        2                   sin(x)                           2                             3                                  sin(x)                             sin(x)                            2                  |    
\                                      sin (x)                                              sin (x)                       sin (x)                                                                                                 sin (x)               /    
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                            sin(x)

$$\frac{\left(- \frac{\left(1 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{\sin{\left(x \right)}} - \frac{\left(1 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \left(1 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(1 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)}{\sin{\left(x \right)}} - \frac{3 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{\sin{\left(x \right)}} - \frac{4 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{3 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + 2 + \frac{4 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) e^{- x}}{\sin{\left(x \right)}}$$

Simplify

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Identical expressions

Derivative of 1/(e^x*sinx)

The graph:

Piecewise:

The solution