Sr Examen

Otras calculadoras


1/(x^3*x^4*cos(x))

Derivada de 1/(x^3*x^4*cos(x))

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
     1      
------------
 3  4       
x *x *cos(x)
1x3x4cos(x)\frac{1}{x^{3} x^{4} \cos{\left(x \right)}}
1/((x^3*x^4)*cos(x))
Solución detallada
  1. Sustituimos u=x3x4cos(x)u = x^{3} x^{4} \cos{\left(x \right)}.

  2. Según el principio, aplicamos: 1u\frac{1}{u} tenemos 1u2- \frac{1}{u^{2}}

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxx3x4cos(x)\frac{d}{d x} x^{3} x^{4} \cos{\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)=x3x4f{\left(x \right)} = x^{3} x^{4}; calculamos 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)=x3f{\left(x \right)} = x^{3}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Según el principio, aplicamos: x3x^{3} tenemos 3x23 x^{2}

        g(x)=x4g{\left(x \right)} = x^{4}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. Según el principio, aplicamos: x4x^{4} tenemos 4x34 x^{3}

        Como resultado de: 7x67 x^{6}

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

      1. La derivada del coseno es igual a menos el seno:

        ddxcos(x)=sin(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

      Como resultado de: x7sin(x)+7x6cos(x)- x^{7} \sin{\left(x \right)} + 7 x^{6} \cos{\left(x \right)}

    Como resultado de la secuencia de reglas:

    x7sin(x)+7x6cos(x)x14cos2(x)- \frac{- x^{7} \sin{\left(x \right)} + 7 x^{6} \cos{\left(x \right)}}{x^{14} \cos^{2}{\left(x \right)}}

  4. Simplificamos:

    xtan(x)7x8cos(x)\frac{x \tan{\left(x \right)} - 7}{x^{8} \cos{\left(x \right)}}


Respuesta:

xtan(x)7x8cos(x)\frac{x \tan{\left(x \right)} - 7}{x^{8} \cos{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-1000000000500000000
Primera derivada [src]
    1     / 7             6       \
---------*\x *sin(x) - 7*x *cos(x)/
 7                                 
x *cos(x)                          
-----------------------------------
              7                    
             x *cos(x)             
1x7cos(x)(x7sin(x)7x6cos(x))x7cos(x)\frac{\frac{1}{x^{7} \cos{\left(x \right)}} \left(x^{7} \sin{\left(x \right)} - 7 x^{6} \cos{\left(x \right)}\right)}{x^{7} \cos{\left(x \right)}}
Segunda derivada [src]
              2                                                                                                                        
-42*cos(x) + x *cos(x) + 14*x*sin(x)   /  7   sin(x)\                          7*(-7*cos(x) + x*sin(x))   (-7*cos(x) + x*sin(x))*sin(x)
------------------------------------ + |- - + ------|*(-7*cos(x) + x*sin(x)) - ------------------------ + -----------------------------
                 x                     \  x   cos(x)/                                     x                           cos(x)           
---------------------------------------------------------------------------------------------------------------------------------------
                                                                8    2                                                                 
                                                               x *cos (x)                                                              
(xsin(x)7cos(x))(sin(x)cos(x)7x)+(xsin(x)7cos(x))sin(x)cos(x)7(xsin(x)7cos(x))x+x2cos(x)+14xsin(x)42cos(x)xx8cos2(x)\frac{\left(x \sin{\left(x \right)} - 7 \cos{\left(x \right)}\right) \left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{7}{x}\right) + \frac{\left(x \sin{\left(x \right)} - 7 \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{7 \left(x \sin{\left(x \right)} - 7 \cos{\left(x \right)}\right)}{x} + \frac{x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} - 42 \cos{\left(x \right)}}{x}}{x^{8} \cos^{2}{\left(x \right)}}
Tercera derivada [src]
                                                                                                                                                                                                                 /  7   sin(x)\ /              2                     \     /  7   sin(x)\                                                             /  7   sin(x)\                                                                                                                   
                                              /              2               \                 3                             2             /              2                     \                                |- - + ------|*\-42*cos(x) + x *cos(x) + 14*x*sin(x)/   7*|- - + ------|*(-7*cos(x) + x*sin(x))        2                             |- - + ------|*(-7*cos(x) + x*sin(x))*sin(x)                                        /              2                     \       
                                              |    56   2*sin (x)   14*sin(x)|   210*cos(x) + x *sin(x) - 126*x*sin(x) - 21*x *cos(x)   21*\-42*cos(x) + x *cos(x) + 14*x*sin(x)/   105*(-7*cos(x) + x*sin(x))   \  x   cos(x)/                                            \  x   cos(x)/                          3*sin (x)*(-7*cos(x) + x*sin(x))   \  x   cos(x)/                                 28*(-7*cos(x) + x*sin(x))*sin(x)   3*\-42*cos(x) + x *cos(x) + 14*x*sin(x)/*sin(x)
-7*cos(x) + x*sin(x) + (-7*cos(x) + x*sin(x))*|1 + -- + --------- - ---------| - ---------------------------------------------------- - ----------------------------------------- + -------------------------- + ----------------------------------------------------- - --------------------------------------- + -------------------------------- + -------------------------------------------- - -------------------------------- + -----------------------------------------------
                                              |     2       2        x*cos(x)|                             2                                                 2                                   2                                         x                                                x                                     2                                      cos(x)                                  x*cos(x)                                   x*cos(x)                   
                                              \    x     cos (x)             /                            x                                                 x                                   x                                                                                                                              cos (x)                                                                                                                                                 
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                        8    2                                                                                                                                                                                                                                         
                                                                                                                                                                                                                                       x *cos (x)                                                                                                                                                                                                                                      
xsin(x)+(xsin(x)7cos(x))(sin(x)cos(x)7x)sin(x)cos(x)+(xsin(x)7cos(x))(2sin2(x)cos2(x)+114sin(x)xcos(x)+56x2)+3(xsin(x)7cos(x))sin2(x)cos2(x)7cos(x)7(xsin(x)7cos(x))(sin(x)cos(x)7x)x28(xsin(x)7cos(x))sin(x)xcos(x)+(sin(x)cos(x)7x)(x2cos(x)+14xsin(x)42cos(x))x+3(x2cos(x)+14xsin(x)42cos(x))sin(x)xcos(x)+105(xsin(x)7cos(x))x221(x2cos(x)+14xsin(x)42cos(x))x2x3sin(x)21x2cos(x)126xsin(x)+210cos(x)x2x8cos2(x)\frac{x \sin{\left(x \right)} + \frac{\left(x \sin{\left(x \right)} - 7 \cos{\left(x \right)}\right) \left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{7}{x}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \left(x \sin{\left(x \right)} - 7 \cos{\left(x \right)}\right) \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1 - \frac{14 \sin{\left(x \right)}}{x \cos{\left(x \right)}} + \frac{56}{x^{2}}\right) + \frac{3 \left(x \sin{\left(x \right)} - 7 \cos{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 7 \cos{\left(x \right)} - \frac{7 \left(x \sin{\left(x \right)} - 7 \cos{\left(x \right)}\right) \left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{7}{x}\right)}{x} - \frac{28 \left(x \sin{\left(x \right)} - 7 \cos{\left(x \right)}\right) \sin{\left(x \right)}}{x \cos{\left(x \right)}} + \frac{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{7}{x}\right) \left(x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} - 42 \cos{\left(x \right)}\right)}{x} + \frac{3 \left(x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} - 42 \cos{\left(x \right)}\right) \sin{\left(x \right)}}{x \cos{\left(x \right)}} + \frac{105 \left(x \sin{\left(x \right)} - 7 \cos{\left(x \right)}\right)}{x^{2}} - \frac{21 \left(x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} - 42 \cos{\left(x \right)}\right)}{x^{2}} - \frac{x^{3} \sin{\left(x \right)} - 21 x^{2} \cos{\left(x \right)} - 126 x \sin{\left(x \right)} + 210 \cos{\left(x \right)}}{x^{2}}}{x^{8} \cos^{2}{\left(x \right)}}
Gráfico
Derivada de 1/(x^3*x^4*cos(x))