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(√xe^(-5x)^(1/3))*tgx+tg3

Derivada de (√xe^(-5x)^(1/3))*tgx+tg3

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

Ha introducido [src]
       3 ______                
       \/ -5*x                 
  _____                        
\/ x*E         *tan(x) + tan(3)
(ex)5x3tan(x)+tan(3)\left(\sqrt{e x}\right)^{\sqrt[3]{- 5 x}} \tan{\left(x \right)} + \tan{\left(3 \right)}
(sqrt(x*E))^((-5*x)^(1/3))*tan(x) + tan(3)
Solución detallada
  1. diferenciamos (ex)5x3tan(x)+tan(3)\left(\sqrt{e x}\right)^{\sqrt[3]{- 5 x}} \tan{\left(x \right)} + \tan{\left(3 \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)=(ex)5x3f{\left(x \right)} = \left(\sqrt{e x}\right)^{\sqrt[3]{- 5 x}}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

        Perola derivada

        (5x)53x33(log(5x3)+1)\left(- 5 x\right)^{\frac{\sqrt[3]{5} \sqrt[3]{- x}}{3}} \left(\log{\left(\sqrt[3]{- 5 x} \right)} + 1\right)

      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:

        sin2(x)+cos2(x)cos2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      Como resultado de: (ex)53x32(sin2(x)+cos2(x))cos2(x)+(5x)53x33(log(5x3)+1)tan(x)\frac{\left(e x\right)^{\frac{\sqrt[3]{5} \sqrt[3]{- x}}{2}} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \left(- 5 x\right)^{\frac{\sqrt[3]{5} \sqrt[3]{- x}}{3}} \left(\log{\left(\sqrt[3]{- 5 x} \right)} + 1\right) \tan{\left(x \right)}

    2. Sustituimos u=3u = 3.

    3. ddutan(u)=1cos2(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: (ex)53x32(sin2(x)+cos2(x))cos2(x)+(5x)53x33(log(5x3)+1)tan(x)\frac{\left(e x\right)^{\frac{\sqrt[3]{5} \sqrt[3]{- x}}{2}} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \left(- 5 x\right)^{\frac{\sqrt[3]{5} \sqrt[3]{- x}}{3}} \left(\log{\left(\sqrt[3]{- 5 x} \right)} + 1\right) \tan{\left(x \right)}

  2. Simplificamos:

    (ex)53x32+(5x)53x33(log(x)+log(5)+3)sin(2x)6cos2(x)\frac{\left(e x\right)^{\frac{\sqrt[3]{5} \sqrt[3]{- x}}{2}} + \frac{\left(- 5 x\right)^{\frac{\sqrt[3]{5} \sqrt[3]{- x}}{3}} \left(\log{\left(- x \right)} + \log{\left(5 \right)} + 3\right) \sin{\left(2 x \right)}}{6}}{\cos^{2}{\left(x \right)}}


Respuesta:

(ex)53x32+(5x)53x33(log(x)+log(5)+3)sin(2x)6cos2(x)\frac{\left(e x\right)^{\frac{\sqrt[3]{5} \sqrt[3]{- x}}{2}} + \frac{\left(- 5 x\right)^{\frac{\sqrt[3]{5} \sqrt[3]{- x}}{3}} \left(\log{\left(- x \right)} + \log{\left(5 \right)} + 3\right) \sin{\left(2 x \right)}}{6}}{\cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-10100.02-0.02
Primera derivada [src]
     3 ___ 3 ____                      3 ___ 3 ____                                                  
     \/ 5 *\/ -x                       \/ 5 *\/ -x                                                   
     ------------                      ------------ /3 ___ 3 ____   3 ___ 3 ____    /  _____\\       
          2       /       2   \             2       |\/ 5 *\/ -x    \/ 5 *\/ -x *log\\/ x*E /|       
(x*E)            *\1 + tan (x)/ + (x*E)            *|------------ + -------------------------|*tan(x)
                                                    \    2*x                   3*x           /       
(ex)53x32(53x3log(ex)3x+53x32x)tan(x)+(ex)53x32(tan2(x)+1)\left(e x\right)^{\frac{\sqrt[3]{5} \sqrt[3]{- x}}{2}} \left(\frac{\sqrt[3]{5} \sqrt[3]{- x} \log{\left(\sqrt{e x} \right)}}{3 x} + \frac{\sqrt[3]{5} \sqrt[3]{- x}}{2 x}\right) \tan{\left(x \right)} + \left(e x\right)^{\frac{\sqrt[3]{5} \sqrt[3]{- x}}{2}} \left(\tan^{2}{\left(x \right)} + 1\right)
Segunda derivada [src]
     3 ___ 3 ____                                                                                                                                                                                                                                     
     \/ 5 *\/ -x                                                                                                                                                                                                                                      
     ------------ /                         3 ___ 3 ____ /         /  ___  1/2\\          3 ___ 3 ____ /       2   \ /         /  ___  1/2\\   3 ___ 3 ____ /       2   \                   2/3     2/3 /         /  ___  1/2\\                      \
          2       |  /       2   \          \/ 5 *\/ -x *\3 + 4*log\\/ x *e   //*tan(x)   \/ 5 *\/ -x *\1 + tan (x)/*\3 + 2*log\\/ x *e   //   \/ 5 *\/ -x *\1 + tan (x)/*(3 + log(E*x))   5   *(-x)   *\3 + 2*log\\/ x *e   //*(3 + log(E*x))*tan(x)|
(E*x)            *|2*\1 + tan (x)/*tan(x) - ------------------------------------------- + -------------------------------------------------- + ----------------------------------------- + ----------------------------------------------------------|
                  |                                                2                                             6*x                                              6*x                                                    2                           |
                  \                                            18*x                                                                                                                                                  36*x                            /
(ex)53x32(2(tan2(x)+1)tan(x)+53x3(log(ex)+3)(tan2(x)+1)6x+53x3(2log(xe12)+3)(tan2(x)+1)6x+523(x)23(log(ex)+3)(2log(xe12)+3)tan(x)36x253x3(4log(xe12)+3)tan(x)18x2)\left(e x\right)^{\frac{\sqrt[3]{5} \sqrt[3]{- x}}{2}} \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \frac{\sqrt[3]{5} \sqrt[3]{- x} \left(\log{\left(e x \right)} + 3\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{6 x} + \frac{\sqrt[3]{5} \sqrt[3]{- x} \left(2 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 3\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{6 x} + \frac{5^{\frac{2}{3}} \left(- x\right)^{\frac{2}{3}} \left(\log{\left(e x \right)} + 3\right) \left(2 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 3\right) \tan{\left(x \right)}}{36 x^{2}} - \frac{\sqrt[3]{5} \sqrt[3]{- x} \left(4 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 3\right) \tan{\left(x \right)}}{18 x^{2}}\right)
Tercera derivada [src]
     3 ___ 3 ____                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
     \/ 5 *\/ -x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
     ------------ /               2                                             2 /         /  ___  1/2\\          3 ___ 3 ____ /       2   \ /         /  ___  1/2\\   3 ___ 3 ____ /       2   \                     2/3     2/3               2 /       2   \   3 ___ 3 ____ /          /  ___  1/2\\           2/3     2/3 /         /  ___  1/2\\                          2/3     2/3                  /         /  ___  1/2\\          3 ___ 3 ____ /       2   \ /         /  ___  1/2\\           2/3     2/3 /       2   \ /         /  ___  1/2\\                    3 ___ 3 ____ /       2   \                      \
          2       |  /       2   \         2    /       2   \   5*(3 + log(E*x)) *\3 + 2*log\\/ x *e   //*tan(x)   \/ 5 *\/ -x *\1 + tan (x)/*\3 + 4*log\\/ x *e   //   \/ 5 *\/ -x *\1 + tan (x)/*(3 + 2*log(E*x))   5   *(-x)   *(3 + log(E*x)) *\1 + tan (x)/   \/ 5 *\/ -x *\9 + 20*log\\/ x *e   //*tan(x)   5   *(-x)   *\3 + 4*log\\/ x *e   //*(3 + log(E*x))*tan(x)   5   *(-x)   *(3 + 2*log(E*x))*\3 + 2*log\\/ x *e   //*tan(x)   \/ 5 *\/ -x *\1 + tan (x)/*\3 + 2*log\\/ x *e   //*tan(x)   5   *(-x)   *\1 + tan (x)/*\3 + 2*log\\/ x *e   //*(3 + log(E*x))   2*\/ 5 *\/ -x *\1 + tan (x)/*(3 + log(E*x))*tan(x)|
(E*x)            *|2*\1 + tan (x)/  + 4*tan (x)*\1 + tan (x)/ - ------------------------------------------------ - -------------------------------------------------- - ------------------------------------------- + ------------------------------------------ + -------------------------------------------- - ---------------------------------------------------------- - ------------------------------------------------------------ + --------------------------------------------------------- + ----------------------------------------------------------------- + --------------------------------------------------|
                  |                                                                       2                                                  2                                                 2                                            2                                             3                                                     3                                                              3                                                         3*x                                                                2                                                        3*x                        |
                  \                                                                  216*x                                                9*x                                              18*x                                         36*x                                          54*x                                                  54*x                                                          108*x                                                                                                                         18*x                                                                                    /
(ex)53x32(2(tan2(x)+1)2+4(tan2(x)+1)tan2(x)+253x3(log(ex)+3)(tan2(x)+1)tan(x)3x+53x3(2log(xe12)+3)(tan2(x)+1)tan(x)3x+523(x)23(log(ex)+3)2(tan2(x)+1)36x2+523(x)23(log(ex)+3)(2log(xe12)+3)(tan2(x)+1)18x253x3(2log(ex)+3)(tan2(x)+1)18x253x3(4log(xe12)+3)(tan2(x)+1)9x25(log(ex)+3)2(2log(xe12)+3)tan(x)216x2523(x)23(log(ex)+3)(4log(xe12)+3)tan(x)54x3523(x)23(2log(ex)+3)(2log(xe12)+3)tan(x)108x3+53x3(20log(xe12)+9)tan(x)54x3)\left(e x\right)^{\frac{\sqrt[3]{5} \sqrt[3]{- x}}{2}} \left(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{2 \sqrt[3]{5} \sqrt[3]{- x} \left(\log{\left(e x \right)} + 3\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{3 x} + \frac{\sqrt[3]{5} \sqrt[3]{- x} \left(2 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 3\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{3 x} + \frac{5^{\frac{2}{3}} \left(- x\right)^{\frac{2}{3}} \left(\log{\left(e x \right)} + 3\right)^{2} \left(\tan^{2}{\left(x \right)} + 1\right)}{36 x^{2}} + \frac{5^{\frac{2}{3}} \left(- x\right)^{\frac{2}{3}} \left(\log{\left(e x \right)} + 3\right) \left(2 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 3\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{18 x^{2}} - \frac{\sqrt[3]{5} \sqrt[3]{- x} \left(2 \log{\left(e x \right)} + 3\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{18 x^{2}} - \frac{\sqrt[3]{5} \sqrt[3]{- x} \left(4 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 3\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{9 x^{2}} - \frac{5 \left(\log{\left(e x \right)} + 3\right)^{2} \left(2 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 3\right) \tan{\left(x \right)}}{216 x^{2}} - \frac{5^{\frac{2}{3}} \left(- x\right)^{\frac{2}{3}} \left(\log{\left(e x \right)} + 3\right) \left(4 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 3\right) \tan{\left(x \right)}}{54 x^{3}} - \frac{5^{\frac{2}{3}} \left(- x\right)^{\frac{2}{3}} \left(2 \log{\left(e x \right)} + 3\right) \left(2 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 3\right) \tan{\left(x \right)}}{108 x^{3}} + \frac{\sqrt[3]{5} \sqrt[3]{- x} \left(20 \log{\left(\sqrt{x} e^{\frac{1}{2}} \right)} + 9\right) \tan{\left(x \right)}}{54 x^{3}}\right)
Gráfico
Derivada de (√xe^(-5x)^(1/3))*tgx+tg3