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(x*sqrt((x^2)*(10^-14)+1))/((x^2)+(10^7))x*exp(-x)

Derivada de (x*sqrt((x^2)*(10^-14)+1))/((x^2)+(10^7))x*exp(-x)

Función f() - derivada -er orden en el punto
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Gráfico:

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Definida a trozos:

Solución

Ha introducido [src]
     ________________      
    /  2                   
x*\/  x *1.0e-14 + 1     -x
---------------------*x*e  
     2                     
    x  + 10000000          
xx1.01014x2+1x2+10000000exx \frac{x \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}}{x^{2} + 10000000} e^{- x}
(((x*sqrt(x^2*1.0e-14 + 1))/(x^2 + 10000000))*x)*exp(-x)
Solución detallada
  1. Se aplica la regla de la derivada parcial:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

    f(x)=x21.01014x2+1f{\left(x \right)} = x^{2} \sqrt{1.0 \cdot 10^{-14} x^{2} + 1} y g(x)=(x2+10000000)exg{\left(x \right)} = \left(x^{2} + 10000000\right) e^{x}.

    Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Se aplica la regla de la derivada de una multiplicación:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=x2f{\left(x \right)} = x^{2}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Según el principio, aplicamos: x2x^{2} tenemos 2x2 x

      g(x)=1.01014x2+1g{\left(x \right)} = \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Sustituimos u=1.01014x2+1u = 1.0 \cdot 10^{-14} x^{2} + 1.

      2. Según el principio, aplicamos: u\sqrt{u} tenemos 12u\frac{1}{2 \sqrt{u}}

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddx(1.01014x2+1)\frac{d}{d x} \left(1.0 \cdot 10^{-14} x^{2} + 1\right):

        1. diferenciamos 1.01014x2+11.0 \cdot 10^{-14} x^{2} + 1 miembro por miembro:

          1. La derivada de una constante 11 es igual a cero.

          2. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

            1. Según el principio, aplicamos: x2x^{2} tenemos 2x2 x

            Entonces, como resultado: 2.01014x2.0 \cdot 10^{-14} x

          Como resultado de: 2.01014x2.0 \cdot 10^{-14} x

        Como resultado de la secuencia de reglas:

        1.01014x1.01014x2+1\frac{1.0 \cdot 10^{-14} x}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}}

      Como resultado de: 1.01014x31.01014x2+1+2x1.01014x2+1\frac{1.0 \cdot 10^{-14} x^{3}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}} + 2 x \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}

    Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Se aplica la regla de la derivada de una multiplicación:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=x2+10000000f{\left(x \right)} = x^{2} + 10000000; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. diferenciamos x2+10000000x^{2} + 10000000 miembro por miembro:

        1. La derivada de una constante 1000000010000000 es igual a cero.

        2. Según el principio, aplicamos: x2x^{2} tenemos 2x2 x

        Como resultado de: 2x2 x

      g(x)=exg{\left(x \right)} = e^{x}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Derivado exe^{x} es.

      Como resultado de: 2xex+(x2+10000000)ex2 x e^{x} + \left(x^{2} + 10000000\right) e^{x}

    Ahora aplicamos la regla de la derivada de una divesión:

    (x21.01014x2+1(2xex+(x2+10000000)ex)+(x2+10000000)(1.01014x31.01014x2+1+2x1.01014x2+1)ex)e2x(x2+10000000)2\frac{\left(- x^{2} \sqrt{1.0 \cdot 10^{-14} x^{2} + 1} \left(2 x e^{x} + \left(x^{2} + 10000000\right) e^{x}\right) + \left(x^{2} + 10000000\right) \left(\frac{1.0 \cdot 10^{-14} x^{3}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}} + 2 x \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}\right) e^{x}\right) e^{- 2 x}}{\left(x^{2} + 10000000\right)^{2}}

  2. Simplificamos:

    x(x(1.01014x2+1)(x2+2x+10000000)+(3.01014x2+2)(x2+10000000))ex1.01014x2+1(x2+10000000)2\frac{x \left(- x \left(1.0 \cdot 10^{-14} x^{2} + 1\right) \left(x^{2} + 2 x + 10000000\right) + \left(3.0 \cdot 10^{-14} x^{2} + 2\right) \left(x^{2} + 10000000\right)\right) e^{- x}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1} \left(x^{2} + 10000000\right)^{2}}


Respuesta:

x(x(1.01014x2+1)(x2+2x+10000000)+(3.01014x2+2)(x2+10000000))ex1.01014x2+1(x2+10000000)2\frac{x \left(- x \left(1.0 \cdot 10^{-14} x^{2} + 1\right) \left(x^{2} + 2 x + 10000000\right) + \left(3.0 \cdot 10^{-14} x^{2} + 2\right) \left(x^{2} + 10000000\right)\right) e^{- x}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1} \left(x^{2} + 10000000\right)^{2}}

Gráfica
02468-8-6-4-2-10100.5-0.5
Primera derivada [src]
/  /   ________________                 2                               \                        \                                 
|  |  /  2                     1.0e-14*x                                |                        |                                 
|  |\/  x *1.0e-14 + 1  + -------------------                           |                        |                                 
|  |                         ________________           ________________|        ________________|             ________________    
|  |                        /  2                   2   /  2             |       /  2             |        2   /  2               -x
|  |                      \/  x *1.0e-14 + 1    2*x *\/  x *1.0e-14 + 1 |   x*\/  x *1.0e-14 + 1 |  -x   x *\/  x *1.0e-14 + 1 *e  
|x*|----------------------------------------- - ------------------------| + ---------------------|*e   - --------------------------
|  |               2                                               2    |        2               |              2                  
|  |              x  + 10000000                     / 2           \     |       x  + 10000000    |             x  + 10000000       
\  \                                                \x  + 10000000/     /                        /                                 
x21.01014x2+1exx2+10000000+(x(2x21.01014x2+1(x2+10000000)2+1.01014x21.01014x2+1+1.01014x2+1x2+10000000)+x1.01014x2+1x2+10000000)ex- \frac{x^{2} \sqrt{1.0 \cdot 10^{-14} x^{2} + 1} e^{- x}}{x^{2} + 10000000} + \left(x \left(- \frac{2 x^{2} \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}}{\left(x^{2} + 10000000\right)^{2}} + \frac{\frac{1.0 \cdot 10^{-14} x^{2}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}} + \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}}{x^{2} + 10000000}\right) + \frac{x \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}}{x^{2} + 10000000}\right) e^{- x}
Segunda derivada [src]
/                                                    /                              /   ________________                 2    \                                             \                                                                                                                                  \    
|                                                    |                      2       |  /              2         1.0e-14*x     |        ________________ /             2    \|                                                                                                                                  |    
|                                                    |             1.0e-28*x      4*|\/  1 + 1.0e-14*x   + -------------------|       /              2  |          4*x     ||                                                                                                                                  |    
|                                                    |-3.0e-14 + --------------     |                         ________________|   2*\/  1 + 1.0e-14*x  *|-1 + -------------||       /                                                        ________________\                                 ________________|    
|     ________________         ________________      |                        2     |                        /              2 |                         |                 2||       |       ________________                 2          2   /              2 |                 2          2   /              2 |    
|    /              2     2   /              2     2 |           1 + 1.0e-14*x      \                      \/  1 + 1.0e-14*x  /                         \     10000000 + x /|       |      /              2         1.0e-14*x        2*x *\/  1 + 1.0e-14*x  |        2.0e-14*x        4*x *\/  1 + 1.0e-14*x  |  -x
|2*\/  1 + 1.0e-14*x   + x *\/  1 + 1.0e-14*x   - x *|------------------------- + --------------------------------------------- - ------------------------------------------| + 2*x*|- 2*\/  1 + 1.0e-14*x   - ------------------- + ------------------------| + ------------------- - ------------------------|*e  
|                                                    |      ________________                                  2                                             2               |       |                             ________________                    2      |      ________________                    2      |    
|                                                    |     /              2                       10000000 + x                                  10000000 + x                |       |                            /              2         10000000 + x       |     /              2         10000000 + x       |    
\                                                    \   \/  1 + 1.0e-14*x                                                                                                  /       \                          \/  1 + 1.0e-14*x                             /   \/  1 + 1.0e-14*x                             /    
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                                                                                                                                                               2                                                                                                                                                    
                                                                                                                                                   10000000 + x                                                                                                                                                     
(x21.01014x2+14x21.01014x2+1x2+10000000x2(21.01014x2+1(4x2x2+100000001)x2+10000000+4(1.01014x21.01014x2+1+1.01014x2+1)x2+10000000+1.01028x21.01014x2+13.010141.01014x2+1)+2.01014x21.01014x2+1+2x(2x21.01014x2+1x2+100000001.01014x21.01014x2+121.01014x2+1)+21.01014x2+1)exx2+10000000\frac{\left(x^{2} \sqrt{1.0 \cdot 10^{-14} x^{2} + 1} - \frac{4 x^{2} \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}}{x^{2} + 10000000} - x^{2} \left(- \frac{2 \sqrt{1.0 \cdot 10^{-14} x^{2} + 1} \left(\frac{4 x^{2}}{x^{2} + 10000000} - 1\right)}{x^{2} + 10000000} + \frac{4 \left(\frac{1.0 \cdot 10^{-14} x^{2}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}} + \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}\right)}{x^{2} + 10000000} + \frac{\frac{1.0 \cdot 10^{-28} x^{2}}{1.0 \cdot 10^{-14} x^{2} + 1} - 3.0 \cdot 10^{-14}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}}\right) + \frac{2.0 \cdot 10^{-14} x^{2}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}} + 2 x \left(\frac{2 x^{2} \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}}{x^{2} + 10000000} - \frac{1.0 \cdot 10^{-14} x^{2}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}} - 2 \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}\right) + 2 \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}\right) e^{- x}}{x^{2} + 10000000}
Tercera derivada [src]
/                            /                              /                      2  \                                                                                                                                                                                                                                                                                          \                                                                                                                                                                                                                                                                                            \    
|                            |                            2 |             3.0e-42*x   |                                                                                                                                                                                                                                                                                          |                                                                                                                                                                                                                                                                                            |    
|                            |                           x *|-3.0e-28 + --------------|      /   ________________                 2    \                                                                                  /             2    \ /   ________________                 2    \                                                                                       |                                                                                                                  /                              /   ________________                 2    \                                             \                                                  |    
|                            |                     2        |                        2|      |  /              2         1.0e-14*x     |     /                      2  \        ________________ /             2    \     |          4*x     | |  /              2         1.0e-14*x     |            ________________ /             2    \          /                      2  \ |                                                                                                                  |                      2       |  /              2         1.0e-14*x     |        ________________ /             2    \|                                                  |    
|                            |            3.0e-28*x         \           1 + 1.0e-14*x /   12*|\/  1 + 1.0e-14*x   + -------------------|     |             1.0e-28*x   |       /              2  |          4*x     |   6*|-1 + -------------|*|\/  1 + 1.0e-14*x   + -------------------|       2   /              2  |          2*x     |        2 |             1.0e-28*x   | |                                                                                                                  |             1.0e-28*x      4*|\/  1 + 1.0e-14*x   + -------------------|       /              2  |          4*x     ||                                                  |    
|                            |3.0e-14 - -------------- + ------------------------------      |                         ________________|   3*|-3.0e-14 + --------------|   6*\/  1 + 1.0e-14*x  *|-1 + -------------|     |                 2| |                         ________________|   24*x *\/  1 + 1.0e-14*x  *|-1 + -------------|     6*x *|-3.0e-14 + --------------| |                                /                                                        ________________\        |-3.0e-14 + --------------     |                         ________________|   2*\/  1 + 1.0e-14*x  *|-1 + -------------||                                  ________________|    
|       ________________     |                       2                        2              |                        /              2 |     |                        2|                         |                 2|     \     10000000 + x / |                        /              2 |                             |                 2|          |                        2| |         ________________       |       ________________                 2          2   /              2 |        |                        2     |                        /              2 |                         |                 2||                 2           2   /              2 |    
|      /              2      |          1 + 1.0e-14*x            1 + 1.0e-14*x               \                      \/  1 + 1.0e-14*x  /     \           1 + 1.0e-14*x /                         \     10000000 + x /                          \                      \/  1 + 1.0e-14*x  /                             \     10000000 + x /          \           1 + 1.0e-14*x / |    2   /              2        |      /              2         1.0e-14*x        2*x *\/  1 + 1.0e-14*x  |      2 |           1 + 1.0e-14*x      \                      \/  1 + 1.0e-14*x  /                         \     10000000 + x /|        6.0e-14*x        12*x *\/  1 + 1.0e-14*x  |  -x
|- 6*\/  1 + 1.0e-14*x   + x*|--------------------------------------------------------- - ---------------------------------------------- - ----------------------------- + ------------------------------------------ + ------------------------------------------------------------------ - ---------------------------------------------- + -----------------------------------| - x *\/  1 + 1.0e-14*x   - 3*x*|- 2*\/  1 + 1.0e-14*x   - ------------------- + ------------------------| + 3*x *|------------------------- + --------------------------------------------- - ------------------------------------------| - ------------------- + -------------------------|*e  
|                            |                      ________________                                                  2                            ________________                                  2                                                        2                                                            2                     ________________                |                                |                             ________________                    2      |        |      ________________                                  2                                             2               |      ________________                     2      |    
|                            |                     /              2                                       10000000 + x                            /              2                       10000000 + x                                             10000000 + x                                              /            2\                     /              2  /            2\|                                |                            /              2         10000000 + x       |        |     /              2                       10000000 + x                                  10000000 + x                |     /              2          10000000 + x       |    
\                            \                   \/  1 + 1.0e-14*x                                                                              \/  1 + 1.0e-14*x                                                                                                                                           \10000000 + x /                   \/  1 + 1.0e-14*x  *\10000000 + x //                                \                          \/  1 + 1.0e-14*x                             /        \   \/  1 + 1.0e-14*x                                                                                                  /   \/  1 + 1.0e-14*x                              /    
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                                                                                                                                                                                                                                                                                                                                   10000000 + x                                                                                                                                                                                                                                                                                                                                    
(x21.01014x2+1+12x21.01014x2+1x2+10000000+3x2(21.01014x2+1(4x2x2+100000001)x2+10000000+4(1.01014x21.01014x2+1+1.01014x2+1)x2+10000000+1.01028x21.01014x2+13.010141.01014x2+1)6.01014x21.01014x2+13x(2x21.01014x2+1x2+100000001.01014x21.01014x2+121.01014x2+1)+x(24x21.01014x2+1(2x2x2+100000001)(x2+10000000)2+6x2(1.01028x21.01014x2+13.01014)1.01014x2+1(x2+10000000)+61.01014x2+1(4x2x2+100000001)x2+10000000+6(1.01014x21.01014x2+1+1.01014x2+1)(4x2x2+100000001)x2+1000000012(1.01014x21.01014x2+1+1.01014x2+1)x2+100000003(1.01028x21.01014x2+13.01014)1.01014x2+1+x2(3.01042x21.01014x2+13.01028)1.01014x2+13.01028x21.01014x2+1+3.010141.01014x2+1)61.01014x2+1)exx2+10000000\frac{\left(- x^{2} \sqrt{1.0 \cdot 10^{-14} x^{2} + 1} + \frac{12 x^{2} \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}}{x^{2} + 10000000} + 3 x^{2} \left(- \frac{2 \sqrt{1.0 \cdot 10^{-14} x^{2} + 1} \left(\frac{4 x^{2}}{x^{2} + 10000000} - 1\right)}{x^{2} + 10000000} + \frac{4 \left(\frac{1.0 \cdot 10^{-14} x^{2}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}} + \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}\right)}{x^{2} + 10000000} + \frac{\frac{1.0 \cdot 10^{-28} x^{2}}{1.0 \cdot 10^{-14} x^{2} + 1} - 3.0 \cdot 10^{-14}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}}\right) - \frac{6.0 \cdot 10^{-14} x^{2}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}} - 3 x \left(\frac{2 x^{2} \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}}{x^{2} + 10000000} - \frac{1.0 \cdot 10^{-14} x^{2}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}} - 2 \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}\right) + x \left(- \frac{24 x^{2} \sqrt{1.0 \cdot 10^{-14} x^{2} + 1} \left(\frac{2 x^{2}}{x^{2} + 10000000} - 1\right)}{\left(x^{2} + 10000000\right)^{2}} + \frac{6 x^{2} \left(\frac{1.0 \cdot 10^{-28} x^{2}}{1.0 \cdot 10^{-14} x^{2} + 1} - 3.0 \cdot 10^{-14}\right)}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1} \left(x^{2} + 10000000\right)} + \frac{6 \sqrt{1.0 \cdot 10^{-14} x^{2} + 1} \left(\frac{4 x^{2}}{x^{2} + 10000000} - 1\right)}{x^{2} + 10000000} + \frac{6 \left(\frac{1.0 \cdot 10^{-14} x^{2}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}} + \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}\right) \left(\frac{4 x^{2}}{x^{2} + 10000000} - 1\right)}{x^{2} + 10000000} - \frac{12 \left(\frac{1.0 \cdot 10^{-14} x^{2}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}} + \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}\right)}{x^{2} + 10000000} - \frac{3 \left(\frac{1.0 \cdot 10^{-28} x^{2}}{1.0 \cdot 10^{-14} x^{2} + 1} - 3.0 \cdot 10^{-14}\right)}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}} + \frac{\frac{x^{2} \left(\frac{3.0 \cdot 10^{-42} x^{2}}{1.0 \cdot 10^{-14} x^{2} + 1} - 3.0 \cdot 10^{-28}\right)}{1.0 \cdot 10^{-14} x^{2} + 1} - \frac{3.0 \cdot 10^{-28} x^{2}}{1.0 \cdot 10^{-14} x^{2} + 1} + 3.0 \cdot 10^{-14}}{\sqrt{1.0 \cdot 10^{-14} x^{2} + 1}}\right) - 6 \sqrt{1.0 \cdot 10^{-14} x^{2} + 1}\right) e^{- x}}{x^{2} + 10000000}
Gráfico
Derivada de (x*sqrt((x^2)*(10^-14)+1))/((x^2)+(10^7))x*exp(-x)