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x*ln(sqrtx-tgx)
Derivada de la función/ sqrtx/ x-tgx/ x*ln(sqrtx-tgx)

Derivada de x*ln(sqrtx-tgx)

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

Ha introducido [src] 
     /  ___         \
x*log\\/ x  - tan(x)/
xlog⁡(x−tan⁡(x))x \log{\left(\sqrt{x} - \tan{\left(x \right)} \right)} 
x*log(sqrt(x) - tan(x))
Solución detallada

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

    1. Según el principio, aplicamos: xx tenemos 11

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

    1. Sustituimos u=x−tan⁡(x)u = \sqrt{x} - \tan{\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 ddx(x−tan⁡(x))\frac{d}{d x} \left(\sqrt{x} - \tan{\left(x \right)}\right):

      1. diferenciamos x−tan⁡(x)\sqrt{x} - \tan{\left(x \right)} miembro por miembro:

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

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

          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)}}

          Entonces, como resultado: −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)cos⁡2(x)+12x- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{1}{2 \sqrt{x}}

      Como resultado de la secuencia de reglas:

      −sin⁡2(x)+cos⁡2(x)cos⁡2(x)+12xx−tan⁡(x)\frac{- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{1}{2 \sqrt{x}}}{\sqrt{x} - \tan{\left(x \right)}}

    Como resultado de: x(−sin⁡2(x)+cos⁡2(x)cos⁡2(x)+12x)x−tan⁡(x)+log⁡(x−tan⁡(x))\frac{x \left(- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{1}{2 \sqrt{x}}\right)}{\sqrt{x} - \tan{\left(x \right)}} + \log{\left(\sqrt{x} - \tan{\left(x \right)} \right)}

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-200200
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
  /        1         2   \                      
x*|-1 + ------- - tan (x)|                      
  |         ___          |                      
  \     2*\/ x           /      /  ___         \
-------------------------- + log\\/ x  - tan(x)/
        ___                                     
      \/ x  - tan(x)
x(−tan⁡2(x)−1+12x)x−tan⁡(x)+log⁡(x−tan⁡(x))\frac{x \left(- \tan^{2}{\left(x \right)} - 1 + \frac{1}{2 \sqrt{x}}\right)}{\sqrt{x} - \tan{\left(x \right)}} + \log{\left(\sqrt{x} - \tan{\left(x \right)} \right)}
Simplificar
Segunda derivada [src] 
 /                          /                              2                         \\ 
 |                          |       /      1          2   \                          || 
 |                          |       |2 - ----- + 2*tan (x)|                          || 
 |                          |       |      ___            |                          || 
 |                          | 1     \    \/ x             /      /       2   \       || 
 |                        x*|---- + ------------------------ + 8*\1 + tan (x)/*tan(x)|| 
 |                          | 3/2          ___                                       || 
 |      1          2        \x           \/ x  - tan(x)                              /| 
-|2 - ----- + 2*tan (x) + ------------------------------------------------------------| 
 |      ___                                            4                              | 
 \    \/ x                                                                            / 
----------------------------------------------------------------------------------------
                                       ___                                              
                                     \/ x  - tan(x)
−x(8(tan⁡2(x)+1)tan⁡(x)+(2tan⁡2(x)+2−1x)2x−tan⁡(x)+1x32)4+2tan⁡2(x)+2−1xx−tan⁡(x)- \frac{\frac{x \left(8 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \frac{\left(2 \tan^{2}{\left(x \right)} + 2 - \frac{1}{\sqrt{x}}\right)^{2}}{\sqrt{x} - \tan{\left(x \right)}} + \frac{1}{x^{\frac{3}{2}}}\right)}{4} + 2 \tan^{2}{\left(x \right)} + 2 - \frac{1}{\sqrt{x}}}{\sqrt{x} - \tan{\left(x \right)}}
Simplificar
Tercera derivada [src] 
 /                                    /                                                      3                                                                                       \                             \ 
 |                                    |                               /      1          2   \                                 / 1       /       2   \       \ /      1          2   \|                             | 
 |                                    |                             2*|2 - ----- + 2*tan (x)|                               3*|---- + 8*\1 + tan (x)/*tan(x)|*|2 - ----- + 2*tan (x)||                             | 
 |                                    |                         2     |      ___            |                                 | 3/2                         | |      ___            ||                             | 
 |                                    |   3        /       2   \      \    \/ x             /          2    /       2   \     \x                            / \    \/ x             /|                            2| 
 |                                  x*|- ---- + 16*\1 + tan (x)/  + -------------------------- + 32*tan (x)*\1 + tan (x)/ + ---------------------------------------------------------|     /      1          2   \ | 
 |                                    |   5/2                                           2                                                           ___                              |   3*|2 - ----- + 2*tan (x)| | 
 |                                    |  x                              /  ___         \                                                          \/ x  - tan(x)                     |     |      ___            | | 
 |  3        /       2   \            \                                 \\/ x  - tan(x)/                                                                                             /     \    \/ x             / | 
-|------ + 6*\1 + tan (x)/*tan(x) + -------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------| 
 |   3/2                                                                                                    8                                                                                  /  ___         \    | 
 \4*x                                                                                                                                                                                        4*\\/ x  - tan(x)/    / 
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                      ___                                                                                                            
                                                                                                    \/ x  - tan(x)
−x(16(tan⁡2(x)+1)2+32(tan⁡2(x)+1)tan⁡2(x)+3(8(tan⁡2(x)+1)tan⁡(x)+1x32)(2tan⁡2(x)+2−1x)x−tan⁡(x)+2(2tan⁡2(x)+2−1x)3(x−tan⁡(x))2−3x52)8+6(tan⁡2(x)+1)tan⁡(x)+3(2tan⁡2(x)+2−1x)24(x−tan⁡(x))+34x32x−tan⁡(x)- \frac{\frac{x \left(16 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 32 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \frac{3 \left(8 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \frac{1}{x^{\frac{3}{2}}}\right) \left(2 \tan^{2}{\left(x \right)} + 2 - \frac{1}{\sqrt{x}}\right)}{\sqrt{x} - \tan{\left(x \right)}} + \frac{2 \left(2 \tan^{2}{\left(x \right)} + 2 - \frac{1}{\sqrt{x}}\right)^{3}}{\left(\sqrt{x} - \tan{\left(x \right)}\right)^{2}} - \frac{3}{x^{\frac{5}{2}}}\right)}{8} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \frac{3 \left(2 \tan^{2}{\left(x \right)} + 2 - \frac{1}{\sqrt{x}}\right)^{2}}{4 \left(\sqrt{x} - \tan{\left(x \right)}\right)} + \frac{3}{4 x^{\frac{3}{2}}}}{\sqrt{x} - \tan{\left(x \right)}}
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Gráfico
Derivada de x*ln(sqrtx-tgx)
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