Derivada de y=x^(sinx+cosx)

Solución

Ha introducido [src]

 sin(x) + cos(x)
x

$$x^{\sin{\left(x \right)} + \cos{\left(x \right)}}$$

x^(sin(x) + cos(x))

Solución detallada

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

Perola derivada

$\left(\log{\left(\sin{\left(x \right)} + \cos{\left(x \right)} \right)} + 1\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{\sin{\left(x \right)} + \cos{\left(x \right)}}$
Simplificamos:

$\left(\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}\right)^{\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}} \left(\log{\left(\sin{\left(x + \frac{\pi}{4} \right)} \right)} + \frac{\log{\left(2 \right)}}{2} + 1\right)$

Respuesta:

$\left(\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}\right)^{\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}} \left(\log{\left(\sin{\left(x + \frac{\pi}{4} \right)} \right)} + \frac{\log{\left(2 \right)}}{2} + 1\right)$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

 sin(x) + cos(x) /sin(x) + cos(x)                            \
x               *|--------------- + (-sin(x) + cos(x))*log(x)|
                 \       x                                   /

$$x^{\sin{\left(x \right)} + \cos{\left(x \right)}} \left(\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) \log{\left(x \right)} + \frac{\sin{\left(x \right)} + \cos{\left(x \right)}}{x}\right)$$

Simplificar

Segunda derivada [src]

                 /                                             2                                                                    \
 cos(x) + sin(x) |/                            cos(x) + sin(x)\    cos(x) + sin(x)                              2*(-cos(x) + sin(x))|
x               *||(-cos(x) + sin(x))*log(x) - ---------------|  - --------------- - (cos(x) + sin(x))*log(x) - --------------------|
                 |\                                   x       /            2                                             x          |
                 \                                                        x                                                         /

$$x^{\sin{\left(x \right)} + \cos{\left(x \right)}} \left(\left(\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \right)} - \frac{\sin{\left(x \right)} + \cos{\left(x \right)}}{x}\right)^{2} - \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \log{\left(x \right)} - \frac{2 \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)}{x} - \frac{\sin{\left(x \right)} + \cos{\left(x \right)}}{x^{2}}\right)$$

Simplificar

Tercera derivada [src]

                 /                                               3                                                                                                                                                                                                                     \
 cos(x) + sin(x) |  /                            cos(x) + sin(x)\                                3*(cos(x) + sin(x))   2*(cos(x) + sin(x))   3*(-cos(x) + sin(x))     /                            cos(x) + sin(x)\ /cos(x) + sin(x)                              2*(-cos(x) + sin(x))\|
x               *|- |(-cos(x) + sin(x))*log(x) - ---------------|  + (-cos(x) + sin(x))*log(x) - ------------------- + ------------------- + -------------------- + 3*|(-cos(x) + sin(x))*log(x) - ---------------|*|--------------- + (cos(x) + sin(x))*log(x) + --------------------||
                 |  \                                   x       /                                         x                      3                     2              \                                   x       / |        2                                             x          ||
                 \                                                                                                              x                     x                                                             \       x                                                         //

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

Simplificar

Gráfico

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Derivada de y=x^(sinx+cosx)

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