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lntanx
Derivada de la función/ lntanx

Derivada de lntanx

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

Gráfico:

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Solución

Ha introducido [src] 
log(x)*tan(x)
log⁡(x)tan⁡(x)\log{\left(x \right)} \tan{\left(x \right)} 
log(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)=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. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-10102000-1000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
tan(x)   /       2   \       
------ + \1 + tan (x)/*log(x)
  x
(tan⁡2(x)+1)log⁡(x)+tan⁡(x)x\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}
Simplificar
Segunda derivada [src] 
             /       2   \                                
  tan(x)   2*\1 + tan (x)/     /       2   \              
- ------ + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)
     2            x                                       
    x
2(tan⁡2(x)+1)log⁡(x)tan⁡(x)+2(tan⁡2(x)+1)x−tan⁡(x)x22 \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   \       
  3*\1 + tan (x)/   2*tan(x)     /       2   \ /         2   \          6*\1 + tan (x)/*tan(x)
- --------------- + -------- + 2*\1 + tan (x)/*\1 + 3*tan (x)/*log(x) + ----------------------
          2             3                                                         x           
         x             x
2(tan⁡2(x)+1)(3tan⁡2(x)+1)log⁡(x)+6(tan⁡2(x)+1)tan⁡(x)x−3(tan⁡2(x)+1)x2+2tan⁡(x)x32 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \log{\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
Gráfico
Derivada de lntanx
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