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e^x*log(sin(x))
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e^x*log(sinx)

The derivative of the function/ e^x*log(sin(x))

Derivative of e^x*log(sin(x))

The solution

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 x            
E *log(sin(x))

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

E^x*log(sin(x))

Detail solution

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.
$g{\left(x \right)} = \log{\left(\sin{\left(x \right)} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
The result is: $e^{x} \log{\left(\sin{\left(x \right)} \right)} + \frac{e^{x} \cos{\left(x \right)}}{\sin{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                         x
 x               cos(x)*e 
e *log(sin(x)) + ---------
                   sin(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of e^x*log(sin(x))

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The solution