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log3(sin²x-e^(2x- one))
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The derivative of the function/ log3(sin²x-e^(2x-1))

Derivative of log3(sin²x-e^(2x-1))

The solution

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

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

  /   /   2       2*x - 1\\
d |log\sin (x) - e       /|
--|-----------------------|
dx\         log(3)        /

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

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Detail solution

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

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

The answer is:

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

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of log3(sin²x-e^(2x-1))

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