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Similar expressions
e^(2x^2)*ln(sinx)

The derivative of the function/ e^(2x^2)*ln(sin(x))

Derivative of e^(2x^2)*ln(sin(x))

The solution

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    2            
 2*x             
e    *log(sin(x))

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

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

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

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = e^{2 x^{2}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x^{2}$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x^{2}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $4 x$
  The result of the chain rule is:
  
  $4 x e^{2 x^{2}}$
$g{\left(x \right)} = \log{\left(\sin{\left(x \right)} \right)}$ ; 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: $4 x e^{2 x^{2}} \log{\left(\sin{\left(x \right)} \right)} + \frac{e^{2 x^{2}} \cos{\left(x \right)}}{\sin{\left(x \right)}}$
Now simplify:

$\left(4 x \log{\left(\sin{\left(x \right)} \right)} + \frac{1}{\tan{\left(x \right)}}\right) e^{2 x^{2}}$

The answer is:

$\left(4 x \log{\left(\sin{\left(x \right)} \right)} + \frac{1}{\tan{\left(x \right)}}\right) e^{2 x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           2                        
        2*x            2            
cos(x)*e            2*x             
------------ + 4*x*e    *log(sin(x))
   sin(x)

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

Simplify

The second derivative [src]

/        2                                           \     2
|     cos (x)     /       2\               8*x*cos(x)|  2*x 
|-1 - ------- + 4*\1 + 4*x /*log(sin(x)) + ----------|*e    
|        2                                   sin(x)  |      
\     sin (x)                                        /

$$\left(4 \cdot \left(4 x^{2} + 1\right) \log{\left(\sin{\left(x \right)} \right)} + \frac{8 x \cos{\left(x \right)}}{\sin{\left(x \right)}} - 1 - \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) e^{2 x^{2}}$$

Simplify

The third derivative [src]

  /                      /       2   \                                                          \      
  |                      |    cos (x)|                                                          |      
  |                      |1 + -------|*cos(x)                                                   |      
  |      /       2   \   |       2   |            /       2\                                    |     2
  |      |    cos (x)|   \    sin (x)/          6*\1 + 4*x /*cos(x)       /       2\            |  2*x 
2*|- 6*x*|1 + -------| + -------------------- + ------------------- + 8*x*\3 + 4*x /*log(sin(x))|*e    
  |      |       2   |          sin(x)                 sin(x)                                   |      
  \      \    sin (x)/                                                                          /

$$2 \left(8 x \left(4 x^{2} + 3\right) \log{\left(\sin{\left(x \right)} \right)} - 6 x \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) + \frac{\left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{6 \cdot \left(4 x^{2} + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) e^{2 x^{2}}$$

Simplify

The graph

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Derivative of e^(2x^2)*ln(sin(x))

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Piecewise:

The solution