Derivative of e^x/sinx

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   x  
  E   
------
sin(x)

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

E^x/sin(x)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = e^{x}$ and $g{\left(x \right)} = \sin{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $e^{x}$ is itself.
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   x             x
  e      cos(x)*e 
------ - ---------
sin(x)       2    
          sin (x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/                           /         2   \       \   
|                           |    6*cos (x)|       |   
|                           |5 + ---------|*cos(x)|   
|                    2      |        2    |       |   
|    3*cos(x)   6*cos (x)   \     sin (x) /       |  x
|4 - -------- + --------- - ----------------------|*e 
|     sin(x)        2               sin(x)        |   
\                sin (x)                          /   
------------------------------------------------------
                        sin(x)

$$\frac{\left(- \frac{\left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 4 - \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) e^{x}}{\sin{\left(x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of e^x/sinx

The graph:

Piecewise:

The solution