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y=√xe^(-5x)*tgx+tg3
Derivada de la función/ e^(-5x)/ √xe^(-5x)*tgx+tg3/ y=√xe^(-5x)*tgx+tg3

Derivada de y=√xe^(-5x)*tgx+tg3

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

Ha introducido [src] 
       -5*x                
  _____                    
\/ x*E     *tan(x) + tan(3)
tan⁡(3)+(ex)−5xtan⁡(x)\tan{\left(3 \right)} + \left(\sqrt{e x}\right)^{- 5 x} \tan{\left(x \right)} 
(sqrt(x*E))^(-5*x)*tan(x) + tan(3)
Solución detallada

  1. diferenciamos tan⁡(3)+(ex)−5xtan⁡(x)\tan{\left(3 \right)} + \left(\sqrt{e x}\right)^{- 5 x} \tan{\left(x \right)} miembro por miembro:

    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)=tan⁡(x)f{\left(x \right)} = \tan{\left(x \right)} y g(x)=(xe12)5xg{\left(x \right)} = \left(\sqrt{x} e^{\frac{1}{2}}\right)^{5 x}.

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

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

      1. No logro encontrar los pasos en la búsqueda de esta derivada.

        Perola derivada

        (5x)5x(log⁡(5x)+1)\left(5 x\right)^{5 x} \left(\log{\left(5 x \right)} + 1\right)

      Ahora aplicamos la regla de la derivada de una divesión:

      (xe12)−10x(−(5x)5x(log⁡(5x)+1)tan⁡(x)+(xe12)5x(sin⁡2(x)+cos⁡2(x))cos⁡2(x))\left(\sqrt{x} e^{\frac{1}{2}}\right)^{- 10 x} \left(- \left(5 x\right)^{5 x} \left(\log{\left(5 x \right)} + 1\right) \tan{\left(x \right)} + \frac{\left(\sqrt{x} e^{\frac{1}{2}}\right)^{5 x} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}\right)

    2. Sustituimos u=3u = 3.

    3. ddutan⁡(u)=1cos⁡2(u)\frac{d}{d u} \tan{\left(u \right)} = \frac{1}{\cos^{2}{\left(u \right)}}

    4. Luego se aplica una cadena de reglas. Multiplicamos por ddx3\frac{d}{d x} 3:

      1. La derivada de una constante 33 es igual a cero.

      Como resultado de la secuencia de reglas:

      00

    Como resultado de: (xe12)−10x(−(5x)5x(log⁡(5x)+1)tan⁡(x)+(xe12)5x(sin⁡2(x)+cos⁡2(x))cos⁡2(x))\left(\sqrt{x} e^{\frac{1}{2}}\right)^{- 10 x} \left(- \left(5 x\right)^{5 x} \left(\log{\left(5 x \right)} + 1\right) \tan{\left(x \right)} + \frac{\left(\sqrt{x} e^{\frac{1}{2}}\right)^{5 x} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}\right)

  2. Simplificamos:

    (xe12)−10x(−(5x)5x(log⁡(5x)+1)sin⁡(2x)2+(xe12)5x)cos⁡2(x)\frac{\left(\sqrt{x} e^{\frac{1}{2}}\right)^{- 10 x} \left(- \frac{\left(5 x\right)^{5 x} \left(\log{\left(5 x \right)} + 1\right) \sin{\left(2 x \right)}}{2} + \left(\sqrt{x} e^{\frac{1}{2}}\right)^{5 x}\right)}{\cos^{2}{\left(x \right)}}

Respuesta:

(xe12)−10x(−(5x)5x(log⁡(5x)+1)sin⁡(2x)2+(xe12)5x)cos⁡2(x)\frac{\left(\sqrt{x} e^{\frac{1}{2}}\right)^{- 10 x} \left(- \frac{\left(5 x\right)^{5 x} \left(\log{\left(5 x \right)} + 1\right) \sin{\left(2 x \right)}}{2} + \left(\sqrt{x} e^{\frac{1}{2}}\right)^{5 x}\right)}{\cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-5e355e35
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Primera derivada [src] 
     -5*x                      -5*x                              
     ----                      ----                              
      2   /       2   \         2   /  5        /  _____\\       
(x*E)    *\1 + tan (x)/ + (x*E)    *|- - - 5*log\\/ x*E /|*tan(x)
                                    \  2                 /
(ex)−5x2(−5log⁡(ex)−52)tan⁡(x)+(ex)−5x2(tan⁡2(x)+1)\left(e x\right)^{- \frac{5 x}{2}} \left(- 5 \log{\left(\sqrt{e x} \right)} - \frac{5}{2}\right) \tan{\left(x \right)} + \left(e x\right)^{- \frac{5 x}{2}} \left(\tan^{2}{\left(x \right)} + 1\right)
Simplificar
Segunda derivada [src] 
     -5*x                                                                                                                                                                  
     ---- /                                      /       2   \ /         /  ___  1/2\\     /       2   \                     /         /  ___  1/2\\                      \
      2   |  /       2   \          5*tan(x)   5*\1 + tan (x)/*\1 + 2*log\\/ x *e   //   5*\1 + tan (x)/*(1 + log(E*x))   25*\1 + 2*log\\/ x *e   //*(1 + log(E*x))*tan(x)|
(E*x)    *|2*\1 + tan (x)/*tan(x) - -------- - --------------------------------------- - ------------------------------ + ------------------------------------------------|
          \                           2*x                         2                                    2                                         4                        /
(ex)−5x2(25(log⁡(ex)+1)(2log⁡(xe12)+1)tan⁡(x)4−5(log⁡(ex)+1)(tan⁡2(x)+1)2−5(2log⁡(xe12)+1)(tan⁡2(x)+1)2+2(tan⁡2(x)+1)tan⁡(x)−5tan⁡(x)2x)\left(e x\right)^{- \frac{5 x}{2}} \left(\frac{25 \left(\log{\left(e x \right)} + 1\right) \left(2 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 1\right) \tan{\left(x \right)}}{4} - \frac{5 \left(\log{\left(e x \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{2} - \frac{5 \left(2 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{5 \tan{\left(x \right)}}{2 x}\right)
Simplificar
Tercera derivada [src] 
     -5*x                                                                                                                                                                                                                                                                                                                                                                                      
     ---- /               2                                /       2   \                               2 /       2   \                                                                                                               2 /         /  ___  1/2\\                                        /       2   \ /         /  ___  1/2\\                     /         /  ___  1/2\\       \
      2   |  /       2   \         2    /       2   \   15*\1 + tan (x)/   5*tan(x)   25*(1 + log(E*x)) *\1 + tan (x)/      /       2   \                           /       2   \ /         /  ___  1/2\\          125*(1 + log(E*x)) *\1 + 2*log\\/ x *e   //*tan(x)   25*(1 + log(E*x))*tan(x)   25*\1 + tan (x)/*\1 + 2*log\\/ x *e   //*(1 + log(E*x))   25*\1 + 2*log\\/ x *e   //*tan(x)|
(E*x)    *|2*\1 + tan (x)/  + 4*tan (x)*\1 + tan (x)/ - ---------------- + -------- + -------------------------------- - 10*\1 + tan (x)/*(1 + log(E*x))*tan(x) - 5*\1 + tan (x)/*\1 + 2*log\\/ x *e   //*tan(x) - -------------------------------------------------- + ------------------------ + ------------------------------------------------------- + ---------------------------------|
          |                                                   2*x               2                    4                                                                                                                                     8                                      2*x                                         2                                             4*x               |
          \                                                                  2*x                                                                                                                                                                                                                                                                                                              /
(ex)−5x2(−125(log⁡(ex)+1)2(2log⁡(xe12)+1)tan⁡(x)8+25(log⁡(ex)+1)2(tan⁡2(x)+1)4+25(log⁡(ex)+1)(2log⁡(xe12)+1)(tan⁡2(x)+1)2−10(log⁡(ex)+1)(tan⁡2(x)+1)tan⁡(x)−5(2log⁡(xe12)+1)(tan⁡2(x)+1)tan⁡(x)+2(tan⁡2(x)+1)2+4(tan⁡2(x)+1)tan⁡2(x)+25(log⁡(ex)+1)tan⁡(x)2x+25(2log⁡(xe12)+1)tan⁡(x)4x−15(tan⁡2(x)+1)2x+5tan⁡(x)2x2)\left(e x\right)^{- \frac{5 x}{2}} \left(- \frac{125 \left(\log{\left(e x \right)} + 1\right)^{2} \left(2 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 1\right) \tan{\left(x \right)}}{8} + \frac{25 \left(\log{\left(e x \right)} + 1\right)^{2} \left(\tan^{2}{\left(x \right)} + 1\right)}{4} + \frac{25 \left(\log{\left(e x \right)} + 1\right) \left(2 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{2} - 10 \left(\log{\left(e x \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - 5 \left(2 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \frac{25 \left(\log{\left(e x \right)} + 1\right) \tan{\left(x \right)}}{2 x} + \frac{25 \left(2 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 1\right) \tan{\left(x \right)}}{4 x} - \frac{15 \left(\tan^{2}{\left(x \right)} + 1\right)}{2 x} + \frac{5 \tan{\left(x \right)}}{2 x^{2}}\right)
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Gráfico
Derivada de y=√xe^(-5x)*tgx+tg3
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