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y=ln(sinx)/5√x^3

Derivada de y=ln(sinx)/5√x^3

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

Ha introducido [src]
                 3
log(sin(x))   ___ 
-----------*\/ x  
     5            
log(sin(x))5(x)3\frac{\log{\left(\sin{\left(x \right)} \right)}}{5} \left(\sqrt{x}\right)^{3}
(log(sin(x))/5)*(sqrt(x))^3
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)=x32log(sin(x))f{\left(x \right)} = x^{\frac{3}{2}} \log{\left(\sin{\left(x \right)} \right)} y g(x)=5g{\left(x \right)} = 5.

    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)=x32f{\left(x \right)} = x^{\frac{3}{2}}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Según el principio, aplicamos: x32x^{\frac{3}{2}} tenemos 3x2\frac{3 \sqrt{x}}{2}

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

      1. Sustituimos u=sin(x)u = \sin{\left(x \right)}.

      2. Derivado log(u)\log{\left(u \right)} es 1u\frac{1}{u}.

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddxsin(x)\frac{d}{d x} \sin{\left(x \right)}:

        1. La derivada del seno es igual al coseno:

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

        Como resultado de la secuencia de reglas:

        cos(x)sin(x)\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}

      Como resultado de: x32cos(x)sin(x)+3xlog(sin(x))2\frac{x^{\frac{3}{2}} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{3 \sqrt{x} \log{\left(\sin{\left(x \right)} \right)}}{2}

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

    1. La derivada de una constante 55 es igual a cero.

    Ahora aplicamos la regla de la derivada de una divesión:

    x32cos(x)5sin(x)+3xlog(sin(x))10\frac{x^{\frac{3}{2}} \cos{\left(x \right)}}{5 \sin{\left(x \right)}} + \frac{3 \sqrt{x} \log{\left(\sin{\left(x \right)} \right)}}{10}

  2. Simplificamos:

    x(2xtan(x)+3log(sin(x)))10\frac{\sqrt{x} \left(\frac{2 x}{\tan{\left(x \right)}} + 3 \log{\left(\sin{\left(x \right)} \right)}\right)}{10}


Respuesta:

x(2xtan(x)+3log(sin(x)))10\frac{\sqrt{x} \left(\frac{2 x}{\tan{\left(x \right)}} + 3 \log{\left(\sin{\left(x \right)} \right)}\right)}{10}

Gráfica
02468-8-6-4-2-1010-250250
Primera derivada [src]
    ___                3/2       
3*\/ x *log(sin(x))   x   *cos(x)
------------------- + -----------
         10             5*sin(x) 
x32cos(x)5sin(x)+3xlog(sin(x))10\frac{x^{\frac{3}{2}} \cos{\left(x \right)}}{5 \sin{\left(x \right)}} + \frac{3 \sqrt{x} \log{\left(\sin{\left(x \right)} \right)}}{10}
Segunda derivada [src]
         /       2   \                        ___       
     3/2 |    cos (x)|   3*log(sin(x))   12*\/ x *cos(x)
- 4*x   *|1 + -------| + ------------- + ---------------
         |       2   |         ___            sin(x)    
         \    sin (x)/       \/ x                       
--------------------------------------------------------
                           20                           
4x32(1+cos2(x)sin2(x))+12xcos(x)sin(x)+3log(sin(x))x20\frac{- 4 x^{\frac{3}{2}} \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) + \frac{12 \sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{3 \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x}}}{20}
Tercera derivada [src]
                                                                  /       2   \       
                                                              3/2 |    cos (x)|       
                                                          16*x   *|1 + -------|*cos(x)
           /       2   \                                          |       2   |       
       ___ |    cos (x)|   3*log(sin(x))    18*cos(x)             \    sin (x)/       
- 36*\/ x *|1 + -------| - ------------- + ------------ + ----------------------------
           |       2   |         3/2         ___                     sin(x)           
           \    sin (x)/        x          \/ x *sin(x)                               
--------------------------------------------------------------------------------------
                                          40                                          
16x32(1+cos2(x)sin2(x))cos(x)sin(x)36x(1+cos2(x)sin2(x))+18cos(x)xsin(x)3log(sin(x))x3240\frac{\frac{16 x^{\frac{3}{2}} \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 36 \sqrt{x} \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) + \frac{18 \cos{\left(x \right)}}{\sqrt{x} \sin{\left(x \right)}} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}}{x^{\frac{3}{2}}}}{40}
Gráfico
Derivada de y=ln(sinx)/5√x^3