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Derivada de la función/ xln|x+(x^2+3)^(1/2)|-(x^2+3)^(1/2)

Derivada de xln|x+(x^2+3)^(1/2)|-(x^2+3)^(1/2)

Función f() - derivada -er orden en el punto
v

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
     /|       ________|\      ________
     ||      /  2     ||     /  2     
x*log\|x + \/  x  + 3 |/ - \/  x  + 3
xlog⁡(∣x+x2+3∣)−x2+3x \log{\left(\left|{x + \sqrt{x^{2} + 3}}\right| \right)} - \sqrt{x^{2} + 3} 
x*log(Abs(x + sqrt(x^2 + 3))) - sqrt(x^2 + 3)
Primera derivada [src] 
                                        /       ________\                         
                  /         x     \     |      /  2     |                         
                x*|1 + -----------|*sign\x + \/  x  + 3 /                         
                  |       ________|                                               
                  |      /  2     |                            /|       ________|\
       x          \    \/  x  + 3 /                            ||      /  2     ||
- ----------- + ----------------------------------------- + log\|x + \/  x  + 3 |/
     ________               |       ________|                                     
    /  2                    |      /  2     |                                     
  \/  x  + 3                |x + \/  x  + 3 |
x(xx2+3+1)sign⁡(x+x2+3)∣x+x2+3∣−xx2+3+log⁡(∣x+x2+3∣)\frac{x \left(\frac{x}{\sqrt{x^{2} + 3}} + 1\right) \operatorname{sign}{\left(x + \sqrt{x^{2} + 3} \right)}}{\left|{x + \sqrt{x^{2} + 3}}\right|} - \frac{x}{\sqrt{x^{2} + 3}} + \log{\left(\left|{x + \sqrt{x^{2} + 3}}\right| \right)}
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Segunda derivada [src] 
                                                      /       ________\                      2      /       ________\                        2           /       ________\                                        
                                /         x     \     |      /      2 |     /         x     \      2|      /      2 |       /         x     \            |      /      2 |     /        2  \     /       ________\
                              2*|1 + -----------|*sign\x + \/  3 + x  /   x*|1 + -----------| *sign \x + \/  3 + x  /   2*x*|1 + -----------| *DiracDelta\x + \/  3 + x  /     |       x   |     |      /      2 |
                                |       ________|                           |       ________|                               |       ________|                                x*|-1 + ------|*sign\x + \/  3 + x  /
                      2         |      /      2 |                           |      /      2 |                               |      /      2 |                                  |          2|                      
       1             x          \    \/  3 + x  /                           \    \/  3 + x  /                               \    \/  3 + x  /                                  \     3 + x /                      
- ----------- + ----------- + ----------------------------------------- - ------------------------------------------- + -------------------------------------------------- - -------------------------------------
     ________           3/2               |       ________|                                             2                               |       ________|                           ________ |       ________|    
    /      2    /     2\                  |      /      2 |                            /       ________\                                |      /      2 |                          /      2  |      /      2 |    
  \/  3 + x     \3 + x /                  |x + \/  3 + x  |                            |      /      2 |                                |x + \/  3 + x  |                        \/  3 + x  *|x + \/  3 + x  |    
                                                                                       \x + \/  3 + x  /
x2(x2+3)32+2x(xx2+3+1)2δ(x+x2+3)∣x+x2+3∣−x(x2x2+3−1)sign⁡(x+x2+3)x2+3∣x+x2+3∣−x(xx2+3+1)2sign⁡2(x+x2+3)(x+x2+3)2+2(xx2+3+1)sign⁡(x+x2+3)∣x+x2+3∣−1x2+3\frac{x^{2}}{\left(x^{2} + 3\right)^{\frac{3}{2}}} + \frac{2 x \left(\frac{x}{\sqrt{x^{2} + 3}} + 1\right)^{2} \delta\left(x + \sqrt{x^{2} + 3}\right)}{\left|{x + \sqrt{x^{2} + 3}}\right|} - \frac{x \left(\frac{x^{2}}{x^{2} + 3} - 1\right) \operatorname{sign}{\left(x + \sqrt{x^{2} + 3} \right)}}{\sqrt{x^{2} + 3} \left|{x + \sqrt{x^{2} + 3}}\right|} - \frac{x \left(\frac{x}{\sqrt{x^{2} + 3}} + 1\right)^{2} \operatorname{sign}^{2}{\left(x + \sqrt{x^{2} + 3} \right)}}{\left(x + \sqrt{x^{2} + 3}\right)^{2}} + \frac{2 \left(\frac{x}{\sqrt{x^{2} + 3}} + 1\right) \operatorname{sign}{\left(x + \sqrt{x^{2} + 3} \right)}}{\left|{x + \sqrt{x^{2} + 3}}\right|} - \frac{1}{\sqrt{x^{2} + 3}}
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Tercera derivada [src] 
                                                 2      /       ________\                      2           /       ________\                                                                3      /       ________\                        3           /       ________   \                        3           /       ________\     /       ________\                                                                    /        2  \           /       ________\            /       ________\                   /        2  \
                                /         x     \      2|      /      2 |     /         x     \            |      /      2 |     /        2  \     /       ________\       /         x     \      2|      /      2 |       /         x     \            |      /      2    |       /         x     \            |      /      2 |     |      /      2 |        /        2  \     /       ________\       /         x     \ |       x   |           |      /      2 |           2|      /      2 | /         x     \ |       x   |
                              3*|1 + -----------| *sign \x + \/  3 + x  /   6*|1 + -----------| *DiracDelta\x + \/  3 + x  /     |       x   |     |      /      2 |   2*x*|1 + -----------| *sign \x + \/  3 + x  /   2*x*|1 + -----------| *DiracDelta\x + \/  3 + x  , 1/   6*x*|1 + -----------| *DiracDelta\x + \/  3 + x  /*sign\x + \/  3 + x  /      2 |       x   |     |      /      2 |   6*x*|1 + -----------|*|-1 + ------|*DiracDelta\x + \/  3 + x  /   3*x*sign \x + \/  3 + x  /*|1 + -----------|*|-1 + ------|
                                |       ________|                             |       ________|                                3*|-1 + ------|*sign\x + \/  3 + x  /       |       ________|                               |       ________|                                       |       ________|                                                      3*x *|-1 + ------|*sign\x + \/  3 + x  /       |       ________| |          2|                                                          |       ________| |          2|
         3                      |      /      2 |                             |      /      2 |                                  |          2|                             |      /      2 |                               |      /      2 |                                       |      /      2 |                                                           |          2|                             |      /      2 | \     3 + x /                                                          |      /      2 | \     3 + x /
      3*x           3*x         \    \/  3 + x  /                             \    \/  3 + x  /                                  \     3 + x /                             \    \/  3 + x  /                               \    \/  3 + x  /                                       \    \/  3 + x  /                                                           \     3 + x /                             \    \/  3 + x  /                                                                        \    \/  3 + x  /              
- ----------- + ----------- - ------------------------------------------- + ------------------------------------------------ - ------------------------------------- + --------------------------------------------- + ----------------------------------------------------- - ------------------------------------------------------------------------ + ---------------------------------------- - --------------------------------------------------------------- + ----------------------------------------------------------
          5/2           3/2                                 2                              |       ________|                          ________ |       ________|                                      3                                  |       ________|                                                                 2                                           3/2 |       ________|                             ________ |       ________|                                                               2              
  /     2\      /     2\                   /       ________\                               |      /      2 |                         /      2  |      /      2 |                     /       ________\                                   |      /      2 |                                                /       ________\                                    /     2\    |      /      2 |                            /      2  |      /      2 |                                     ________ /       ________\               
  \3 + x /      \3 + x /                   |      /      2 |                               |x + \/  3 + x  |                       \/  3 + x  *|x + \/  3 + x  |                     |      /      2 |                                   |x + \/  3 + x  |                                                |      /      2 |                                    \3 + x /   *|x + \/  3 + x  |                          \/  3 + x  *|x + \/  3 + x  |                                    /      2  |      /      2 |               
                                           \x + \/  3 + x  /                                                                                                                         \x + \/  3 + x  /                                                                                                    \x + \/  3 + x  /                                                                                                                                                          \/  3 + x  *\x + \/  3 + x  /
−3x3(x2+3)52+3x2(x2x2+3−1)sign⁡(x+x2+3)(x2+3)32∣x+x2+3∣+2x(xx2+3+1)3δ(1)(x+x2+3)∣x+x2+3∣−6x(xx2+3+1)(x2x2+3−1)δ(x+x2+3)x2+3∣x+x2+3∣+3x(x2+3)32−6x(xx2+3+1)3δ(x+x2+3)sign⁡(x+x2+3)(x+x2+3)2+3x(xx2+3+1)(x2x2+3−1)sign⁡2(x+x2+3)(x+x2+3)2x2+3+2x(xx2+3+1)3sign⁡2(x+x2+3)(x+x2+3)3+6(xx2+3+1)2δ(x+x2+3)∣x+x2+3∣−3(x2x2+3−1)sign⁡(x+x2+3)x2+3∣x+x2+3∣−3(xx2+3+1)2sign⁡2(x+x2+3)(x+x2+3)2- \frac{3 x^{3}}{\left(x^{2} + 3\right)^{\frac{5}{2}}} + \frac{3 x^{2} \left(\frac{x^{2}}{x^{2} + 3} - 1\right) \operatorname{sign}{\left(x + \sqrt{x^{2} + 3} \right)}}{\left(x^{2} + 3\right)^{\frac{3}{2}} \left|{x + \sqrt{x^{2} + 3}}\right|} + \frac{2 x \left(\frac{x}{\sqrt{x^{2} + 3}} + 1\right)^{3} \delta^{\left( 1 \right)}\left( x + \sqrt{x^{2} + 3} \right)}{\left|{x + \sqrt{x^{2} + 3}}\right|} - \frac{6 x \left(\frac{x}{\sqrt{x^{2} + 3}} + 1\right) \left(\frac{x^{2}}{x^{2} + 3} - 1\right) \delta\left(x + \sqrt{x^{2} + 3}\right)}{\sqrt{x^{2} + 3} \left|{x + \sqrt{x^{2} + 3}}\right|} + \frac{3 x}{\left(x^{2} + 3\right)^{\frac{3}{2}}} - \frac{6 x \left(\frac{x}{\sqrt{x^{2} + 3}} + 1\right)^{3} \delta\left(x + \sqrt{x^{2} + 3}\right) \operatorname{sign}{\left(x + \sqrt{x^{2} + 3} \right)}}{\left(x + \sqrt{x^{2} + 3}\right)^{2}} + \frac{3 x \left(\frac{x}{\sqrt{x^{2} + 3}} + 1\right) \left(\frac{x^{2}}{x^{2} + 3} - 1\right) \operatorname{sign}^{2}{\left(x + \sqrt{x^{2} + 3} \right)}}{\left(x + \sqrt{x^{2} + 3}\right)^{2} \sqrt{x^{2} + 3}} + \frac{2 x \left(\frac{x}{\sqrt{x^{2} + 3}} + 1\right)^{3} \operatorname{sign}^{2}{\left(x + \sqrt{x^{2} + 3} \right)}}{\left(x + \sqrt{x^{2} + 3}\right)^{3}} + \frac{6 \left(\frac{x}{\sqrt{x^{2} + 3}} + 1\right)^{2} \delta\left(x + \sqrt{x^{2} + 3}\right)}{\left|{x + \sqrt{x^{2} + 3}}\right|} - \frac{3 \left(\frac{x^{2}}{x^{2} + 3} - 1\right) \operatorname{sign}{\left(x + \sqrt{x^{2} + 3} \right)}}{\sqrt{x^{2} + 3} \left|{x + \sqrt{x^{2} + 3}}\right|} - \frac{3 \left(\frac{x}{\sqrt{x^{2} + 3}} + 1\right)^{2} \operatorname{sign}^{2}{\left(x + \sqrt{x^{2} + 3} \right)}}{\left(x + \sqrt{x^{2} + 3}\right)^{2}}
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