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Derivada de la función/ y=log/ xtanx/ 10^10/ y=logxtanx+10^10^x

Derivada de y=logxtanx+10^10^x

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

Ha introducido [src] 
                  /  x\
                  \10 /
log(x)*tan(x) + 10
1010x+log⁡(x)tan⁡(x)10^{10^{x}} + \log{\left(x \right)} \tan{\left(x \right)} 
log(x)*tan(x) + 10^(10^x)
Solución detallada

  1. diferenciamos 1010x+log⁡(x)tan⁡(x)10^{10^{x}} + \log{\left(x \right)} \tan{\left(x \right)} miembro por miembro:

    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)=log⁡(x)f{\left(x \right)} = \log{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

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

      1. Reescribimos las funciones para diferenciar:

        tan⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

      2. 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)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} y g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

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

        Para calcular 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)}

        Ahora aplicamos la regla de la derivada de una divesión:

        sin⁡2(x)+cos⁡2(x)cos⁡2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      Como resultado de: (sin⁡2(x)+cos⁡2(x))log⁡(x)cos⁡2(x)+tan⁡(x)x\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{x}

    2. Sustituimos u=10xu = 10^{x}.

    3. ddu10u=10ulog⁡(10)\frac{d}{d u} 10^{u} = 10^{u} \log{\left(10 \right)}

    4. Luego se aplica una cadena de reglas. Multiplicamos por ddx10x\frac{d}{d x} 10^{x}:

      1. ddx10x=10xlog⁡(10)\frac{d}{d x} 10^{x} = 10^{x} \log{\left(10 \right)}

      Como resultado de la secuencia de reglas:

      1010x10xlog⁡(10)210^{10^{x}} 10^{x} \log{\left(10 \right)}^{2}

    Como resultado de: 1010x10xlog⁡(10)2+(sin⁡2(x)+cos⁡2(x))log⁡(x)cos⁡2(x)+tan⁡(x)x10^{10^{x}} 10^{x} \log{\left(10 \right)}^{2} + \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{x}

  2. Simplificamos:

    1010x+xlog⁡(10)2+log⁡(x)cos⁡2(x)+sin⁡(2x)2xcos⁡2(x)10^{10^{x} + x} \log{\left(10 \right)}^{2} + \frac{\log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(2 x \right)}}{2 x \cos^{2}{\left(x \right)}}

Respuesta:

1010x+xlog⁡(10)2+log⁡(x)cos⁡2(x)+sin⁡(2x)2xcos⁡2(x)10^{10^{x} + x} \log{\left(10 \right)}^{2} + \frac{\log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(2 x \right)}}{2 x \cos^{2}{\left(x \right)}}

Primera derivada [src] 
                                      /  x\         
tan(x)   /       2   \            x   \10 /    2    
------ + \1 + tan (x)/*log(x) + 10 *10     *log (10)
  x
1010x10xlog⁡(10)2+(tan⁡2(x)+1)log⁡(x)+tan⁡(x)x10^{10^{x}} 10^{x} \log{\left(10 \right)}^{2} + \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}
Simplificar
Segunda derivada [src] 
             /       2   \         /  x\              /  x\                                               
  tan(x)   2*\1 + tan (x)/     x   \10 /    3         \10 /   2*x    4         /       2   \              
- ------ + --------------- + 10 *10     *log (10) + 10     *10   *log (10) + 2*\1 + tan (x)/*log(x)*tan(x)
     2            x                                                                                       
    x
1010x102xlog⁡(10)4+1010x10xlog⁡(10)3+2(tan⁡2(x)+1)log⁡(x)tan⁡(x)+2(tan⁡2(x)+1)x−tan⁡(x)x210^{10^{x}} 10^{2 x} \log{\left(10 \right)}^{4} + 10^{10^{x}} 10^{x} \log{\left(10 \right)}^{3} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{\tan{\left(x \right)}}{x^{2}}
Simplificar
Tercera derivada [src] 
    /       2   \                             2                /  x\              /  x\                      /  x\                                                     /       2   \       
  3*\1 + tan (x)/   2*tan(x)     /       2   \             x   \10 /    4         \10 /   3*x    6           \10 /   2*x    5            2    /       2   \          6*\1 + tan (x)/*tan(x)
- --------------- + -------- + 2*\1 + tan (x)/ *log(x) + 10 *10     *log (10) + 10     *10   *log (10) + 3*10     *10   *log (10) + 4*tan (x)*\1 + tan (x)/*log(x) + ----------------------
          2             3                                                                                                                                                      x           
         x             x
1010x103xlog⁡(10)6+3⋅1010x102xlog⁡(10)5+1010x10xlog⁡(10)4+2(tan⁡2(x)+1)2log⁡(x)+4(tan⁡2(x)+1)log⁡(x)tan⁡2(x)+6(tan⁡2(x)+1)tan⁡(x)x−3(tan⁡2(x)+1)x2+2tan⁡(x)x310^{10^{x}} 10^{3 x} \log{\left(10 \right)}^{6} + 3 \cdot 10^{10^{x}} 10^{2 x} \log{\left(10 \right)}^{5} + 10^{10^{x}} 10^{x} \log{\left(10 \right)}^{4} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan^{2}{\left(x \right)} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2}} + \frac{2 \tan{\left(x \right)}}{x^{3}}
Simplificar
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