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y=inx/sinx
Derivada de la función/ y=inx/ x/sinx/ y=inx/sinx

Derivada de y=inx/sinx

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

Gráfico:

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Solución

Ha introducido [src] 
log(x)
------
sin(x)
log⁡(x)sin⁡(x)\frac{\log{\left(x \right)}}{\sin{\left(x \right)}} 
log(x)/sin(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)=log⁡(x)f{\left(x \right)} = \log{\left(x \right)} y g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}.

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

    1. Derivado log⁡(x)\log{\left(x \right)} es 1x\frac{1}{x}.

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

    Ahora aplicamos la regla de la derivada de una divesión:

    −log⁡(x)cos⁡(x)+sin⁡(x)xsin⁡2(x)\frac{- \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}}{\sin^{2}{\left(x \right)}}

  2. Simplificamos:

    −xlog⁡(x)cos⁡(x)+sin⁡(x)xsin⁡2(x)\frac{- x \log{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}}{x \sin^{2}{\left(x \right)}}

Respuesta:

−xlog⁡(x)cos⁡(x)+sin⁡(x)xsin⁡2(x)\frac{- x \log{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}}{x \sin^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-101020000-10000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
   1       cos(x)*log(x)
-------- - -------------
x*sin(x)         2      
              sin (x)
−log⁡(x)cos⁡(x)sin⁡2(x)+1xsin⁡(x)- \frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{x \sin{\left(x \right)}}
Simplificar
Segunda derivada [src] 
       /         2   \                  
  1    |    2*cos (x)|          2*cos(x)
- -- + |1 + ---------|*log(x) - --------
   2   |        2    |          x*sin(x)
  x    \     sin (x) /                  
----------------------------------------
                 sin(x)
(1+2cos⁡2(x)sin⁡2(x))log⁡(x)−2cos⁡(x)xsin⁡(x)−1x2sin⁡(x)\frac{\left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{1}{x^{2}}}{\sin{\left(x \right)}}
Simplificar
Tercera derivada [src] 
       /         2   \               /         2   \              
       |    2*cos (x)|               |    6*cos (x)|              
     3*|1 + ---------|               |5 + ---------|*cos(x)*log(x)
       |        2    |               |        2    |              
2      \     sin (x) /    3*cos(x)   \     sin (x) /              
-- + ----------------- + --------- - -----------------------------
 3           x            2                      sin(x)           
x                        x *sin(x)                                
------------------------------------------------------------------
                              sin(x)
−(5+6cos⁡2(x)sin⁡2(x))log⁡(x)cos⁡(x)sin⁡(x)+3(1+2cos⁡2(x)sin⁡2(x))x+3cos⁡(x)x2sin⁡(x)+2x3sin⁡(x)\frac{- \frac{\left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{3 \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)}{x} + \frac{3 \cos{\left(x \right)}}{x^{2} \sin{\left(x \right)}} + \frac{2}{x^{3}}}{\sin{\left(x \right)}}
Simplificar
Gráfico
Derivada de y=inx/sinx
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