Sr Examen

Otras calculadoras

y=sin*x*tg*x
Derivada de la función/ sin*x/ y=sin/ x*tg*x/ y=sin*x*tg*x

Derivada de y=sin*x*tg*x

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
sin(x)*tan(x)
sin⁡(x)tan⁡(x)\sin{\left(x \right)} \tan{\left(x \right)} 
sin(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)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)}; calculamos 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)}

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

  2. Simplificamos:

    (1+1cos⁡2(x))sin⁡(x)\left(1 + \frac{1}{\cos^{2}{\left(x \right)}}\right) \sin{\left(x \right)}

Respuesta:

(1+1cos⁡2(x))sin⁡(x)\left(1 + \frac{1}{\cos^{2}{\left(x \right)}}\right) \sin{\left(x \right)}

Gráfica
02468-8-6-4-2-1010-20002000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
/       2   \                       
\1 + tan (x)/*sin(x) + cos(x)*tan(x)
(tan⁡2(x)+1)sin⁡(x)+cos⁡(x)tan⁡(x)\left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)}
Simplificar
Segunda derivada [src] 
                   /       2   \            /       2   \              
-sin(x)*tan(x) + 2*\1 + tan (x)/*cos(x) + 2*\1 + tan (x)/*sin(x)*tan(x)
2(tan⁡2(x)+1)sin⁡(x)tan⁡(x)+2(tan⁡2(x)+1)cos⁡(x)−sin⁡(x)tan⁡(x)2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \tan{\left(x \right)}
Simplificar
Tercera derivada [src] 
                   /       2   \            /       2   \ /         2   \            /       2   \              
-cos(x)*tan(x) - 3*\1 + tan (x)/*sin(x) + 2*\1 + tan (x)/*\1 + 3*tan (x)/*sin(x) + 6*\1 + tan (x)/*cos(x)*tan(x)
2(tan⁡2(x)+1)(3tan⁡2(x)+1)sin⁡(x)−3(tan⁡2(x)+1)sin⁡(x)+6(tan⁡2(x)+1)cos⁡(x)tan⁡(x)−cos⁡(x)tan⁡(x)2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} - 3 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)} - \cos{\left(x \right)} \tan{\left(x \right)}
Simplificar
Gráfico
Derivada de y=sin*x*tg*x
    © Sr Examen – Калькулятор