Производная log(Abs(sin(x)))

Вы ввели

log(|sin(x)|)

$$\log{\left(\left|{\sin{\left(x \right)}}\right| \right)}$$

d                
--(log(|sin(x)|))
dx               

$$\frac{d}{d x} \log{\left(\left|{\sin{\left(x \right)}}\right| \right)}$$

Первая производная

cos(x)*sign(sin(x))
-------------------
      |sin(x)|     

$$\frac{\cos{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)}}{\left|{\sin{\left(x \right)}}\right|}$$

Вторая производная

                           2        2                2                      
  sign(sin(x))*sin(x)   cos (x)*sign (sin(x))   2*cos (x)*DiracDelta(sin(x))
- ------------------- - --------------------- + ----------------------------
        |sin(x)|                  2                       |sin(x)|          
                               sin (x)                                      

$$- \frac{\sin{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)}}{\left|{\sin{\left(x \right)}}\right|} + \frac{2 \cos^{2}{\left(x \right)} \delta\left(\sin{\left(x \right)}\right)}{\left|{\sin{\left(x \right)}}\right|} - \frac{\cos^{2}{\left(x \right)} \operatorname{sign}^{2}{\left(\sin{\left(x \right)} \right)}}{\sin^{2}{\left(x \right)}}$$

Третья производная

/                       2                                              2                                 2        2                2                                   \       
|  sign(sin(x))   3*sign (sin(x))   6*DiracDelta(sin(x))*sin(x)   2*cos (x)*DiracDelta(sin(x), 1)   2*cos (x)*sign (sin(x))   6*cos (x)*DiracDelta(sin(x))*sign(sin(x))|       
|- ------------ + --------------- - --------------------------- + ------------------------------- + ----------------------- - -----------------------------------------|*cos(x)
|    |sin(x)|          sin(x)                 |sin(x)|                        |sin(x)|                         3                                  2                    |       
\                                                                                                           sin (x)                            sin (x)                 /       

$$\left(- \frac{6 \sin{\left(x \right)} \delta\left(\sin{\left(x \right)}\right)}{\left|{\sin{\left(x \right)}}\right|} + \frac{2 \cos^{2}{\left(x \right)} \delta^{\left( 1 \right)}\left( \sin{\left(x \right)} \right)}{\left|{\sin{\left(x \right)}}\right|} - \frac{\operatorname{sign}{\left(\sin{\left(x \right)} \right)}}{\left|{\sin{\left(x \right)}}\right|} + \frac{3 \operatorname{sign}^{2}{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}} - \frac{6 \cos^{2}{\left(x \right)} \delta\left(\sin{\left(x \right)}\right) \operatorname{sign}{\left(\sin{\left(x \right)} \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \cos^{2}{\left(x \right)} \operatorname{sign}^{2}{\left(\sin{\left(x \right)} \right)}}{\sin^{3}{\left(x \right)}}\right) \cos{\left(x \right)}$$

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