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y=3^xtgx^2
Derivada de la función/ y=3^x/ tgx^2/ y=3^xtgx^2

Derivada de y=3^xtgx^2

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

Ha introducido [src] 
 x    2   
3 *tan (x)
3xtan⁡2(x)3^{x} \tan^{2}{\left(x \right)} 
3^x*tan(x)^2
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)=3xf{\left(x \right)} = 3^{x}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. ddx3x=3xlog⁡(3)\frac{d}{d x} 3^{x} = 3^{x} \log{\left(3 \right)}

    g(x)=tan⁡2(x)g{\left(x \right)} = \tan^{2}{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Sustituimos u=tan⁡(x)u = \tan{\left(x \right)}.

    2. Según el principio, aplicamos: u2u^{2} tenemos 2u2 u

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan⁡(x)\frac{d}{d x} \tan{\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 la secuencia de reglas:

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

    Como resultado de: 2⋅3x(sin⁡2(x)+cos⁡2(x))tan⁡(x)cos⁡2(x)+3xlog⁡(3)tan⁡2(x)\frac{2 \cdot 3^{x} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 3^{x} \log{\left(3 \right)} \tan^{2}{\left(x \right)}

  2. Simplificamos:

    3x(2sin⁡(x)cos⁡3(x)+log⁡(3)tan⁡2(x))3^{x} \left(\frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \log{\left(3 \right)} \tan^{2}{\left(x \right)}\right)

Respuesta:

3x(2sin⁡(x)cos⁡3(x)+log⁡(3)tan⁡2(x))3^{x} \left(\frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \log{\left(3 \right)} \tan^{2}{\left(x \right)}\right)

Gráfica
02468-8-6-4-2-1010-5000000050000000
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Primera derivada [src] 
 x    2              x /         2   \       
3 *tan (x)*log(3) + 3 *\2 + 2*tan (x)/*tan(x)
3x(2tan⁡2(x)+2)tan⁡(x)+3xlog⁡(3)tan⁡2(x)3^{x} \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} + 3^{x} \log{\left(3 \right)} \tan^{2}{\left(x \right)}
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Segunda derivada [src] 
 x /   2       2        /       2   \ /         2   \     /       2   \              \
3 *\log (3)*tan (x) + 2*\1 + tan (x)/*\1 + 3*tan (x)/ + 4*\1 + tan (x)/*log(3)*tan(x)/
3x(2(tan⁡2(x)+1)(3tan⁡2(x)+1)+4(tan⁡2(x)+1)log⁡(3)tan⁡(x)+log⁡(3)2tan⁡2(x))3^{x} \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)} \tan{\left(x \right)} + \log{\left(3 \right)}^{2} \tan^{2}{\left(x \right)}\right)
Simplificar
Tercera derivada [src] 
 x /   3       2           2    /       2   \            /       2   \ /         2   \            /       2   \ /         2   \       \
3 *\log (3)*tan (x) + 6*log (3)*\1 + tan (x)/*tan(x) + 6*\1 + tan (x)/*\1 + 3*tan (x)/*log(3) + 8*\1 + tan (x)/*\2 + 3*tan (x)/*tan(x)/
3x(6(tan⁡2(x)+1)(3tan⁡2(x)+1)log⁡(3)+8(tan⁡2(x)+1)(3tan⁡2(x)+2)tan⁡(x)+6(tan⁡2(x)+1)log⁡(3)2tan⁡(x)+log⁡(3)3tan⁡2(x))3^{x} \left(6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)} + 8 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)}^{2} \tan{\left(x \right)} + \log{\left(3 \right)}^{3} \tan^{2}{\left(x \right)}\right)
Simplificar
Gráfico
Derivada de y=3^xtgx^2
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