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Derivada de la función/ sinx^2/ arcsin/ y=(5+sinx)^arcsinx^2

Derivada de y=(5+sinx)^arcsinx^2

Solución

Ha introducido [src]

                2   
            asin (x)
(5 + sin(x))

$$\left(\sin{\left(x \right)} + 5\right)^{\operatorname{asin}^{2}{\left(x \right)}}$$

(5 + sin(x))^(asin(x)^2)

Solución detallada

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

Perola derivada

$\left(\log{\left(\operatorname{asin}^{2}{\left(x \right)} \right)} + 1\right) \left(\operatorname{asin}^{2}{\left(x \right)}\right)^{\operatorname{asin}^{2}{\left(x \right)}}$

Respuesta:

$\left(\log{\left(\operatorname{asin}^{2}{\left(x \right)} \right)} + 1\right) \left(\operatorname{asin}^{2}{\left(x \right)}\right)^{\operatorname{asin}^{2}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

                2    /    2                                      \
            asin (x) |asin (x)*cos(x)   2*asin(x)*log(5 + sin(x))|
(5 + sin(x))        *|--------------- + -------------------------|
                     |   5 + sin(x)               ________       |
                     |                           /      2        |
                     \                         \/  1 - x         /

$$\left(\frac{\cos{\left(x \right)} \operatorname{asin}^{2}{\left(x \right)}}{\sin{\left(x \right)} + 5} + \frac{2 \log{\left(\sin{\left(x \right)} + 5 \right)} \operatorname{asin}{\left(x \right)}}{\sqrt{1 - x^{2}}}\right) \left(\sin{\left(x \right)} + 5\right)^{\operatorname{asin}^{2}{\left(x \right)}}$$

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Segunda derivada [src]

                2    /                                    2                                    2                 2       2                                                            \
            asin (x) |/2*log(5 + sin(x))   asin(x)*cos(x)\      2      2*log(5 + sin(x))   asin (x)*sin(x)   asin (x)*cos (x)   2*x*asin(x)*log(5 + sin(x))       4*asin(x)*cos(x)    |
(5 + sin(x))        *||----------------- + --------------| *asin (x) - ----------------- - --------------- - ---------------- + --------------------------- + ------------------------|
                     ||      ________        5 + sin(x)  |                        2           5 + sin(x)                  2                     3/2              ________             |
                     ||     /      2                     |                  -1 + x                            (5 + sin(x))              /     2\                /      2              |
                     \\   \/  1 - x                      /                                                                              \1 - x /              \/  1 - x  *(5 + sin(x))/

$$\left(\sin{\left(x \right)} + 5\right)^{\operatorname{asin}^{2}{\left(x \right)}} \left(\frac{2 x \log{\left(\sin{\left(x \right)} + 5 \right)} \operatorname{asin}{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \left(\frac{\cos{\left(x \right)} \operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)} + 5} + \frac{2 \log{\left(\sin{\left(x \right)} + 5 \right)}}{\sqrt{1 - x^{2}}}\right)^{2} \operatorname{asin}^{2}{\left(x \right)} - \frac{\sin{\left(x \right)} \operatorname{asin}^{2}{\left(x \right)}}{\sin{\left(x \right)} + 5} - \frac{\cos^{2}{\left(x \right)} \operatorname{asin}^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} + 5\right)^{2}} - \frac{2 \log{\left(\sin{\left(x \right)} + 5 \right)}}{x^{2} - 1} + \frac{4 \cos{\left(x \right)} \operatorname{asin}{\left(x \right)}}{\sqrt{1 - x^{2}} \left(\sin{\left(x \right)} + 5\right)}\right)$$

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Tercera derivada [src]

                2    /                                    3                2                                                                             /                        2                 2       2                                                            \                                             2       3                                                                2                        2                       2                                                   \
            asin (x) |/2*log(5 + sin(x))   asin(x)*cos(x)\      3      asin (x)*cos(x)          6*cos(x)            /2*log(5 + sin(x))   asin(x)*cos(x)\ |2*log(5 + sin(x))   asin (x)*sin(x)   asin (x)*cos (x)       4*asin(x)*cos(x)       2*x*asin(x)*log(5 + sin(x))|           2*asin(x)*log(5 + sin(x))   2*asin (x)*cos (x)   6*x*log(5 + sin(x))       6*asin(x)*sin(x)           6*cos (x)*asin(x)       3*asin (x)*cos(x)*sin(x)   6*x *asin(x)*log(5 + sin(x))      6*x*asin(x)*cos(x)   |
(5 + sin(x))        *||----------------- + --------------| *asin (x) - --------------- - ---------------------- - 3*|----------------- + --------------|*|----------------- + --------------- + ---------------- - ------------------------ - ---------------------------|*asin(x) + ------------------------- + ------------------ + ------------------- - ------------------------ - ------------------------- + ------------------------ + ---------------------------- + ------------------------|
                     ||      ________        5 + sin(x)  |                5 + sin(x)     /      2\                  |      ________        5 + sin(x)  | |           2           5 + sin(x)                  2        ________                                3/2        |                          3/2                        3                    2          ________                   ________                                  2                         5/2                    3/2             |
                     ||     /      2                     |                               \-1 + x /*(5 + sin(x))     |     /      2                     | |     -1 + x                            (5 + sin(x))        /      2                         /     2\           |                  /     2\               (5 + sin(x))            /      2\          /      2                   /      2              2        (5 + sin(x))                  /     2\               /     2\                |
                     \\   \/  1 - x                      /                                                          \   \/  1 - x                      / \                                                         \/  1 - x  *(5 + sin(x))           \1 - x /           /                  \1 - x /                                       \-1 + x /        \/  1 - x  *(5 + sin(x))   \/  1 - x  *(5 + sin(x))                                       \1 - x /               \1 - x /   *(5 + sin(x))/

$$\left(\sin{\left(x \right)} + 5\right)^{\operatorname{asin}^{2}{\left(x \right)}} \left(\frac{6 x^{2} \log{\left(\sin{\left(x \right)} + 5 \right)} \operatorname{asin}{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{5}{2}}} + \frac{6 x \log{\left(\sin{\left(x \right)} + 5 \right)}}{\left(x^{2} - 1\right)^{2}} + \frac{6 x \cos{\left(x \right)} \operatorname{asin}{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}} \left(\sin{\left(x \right)} + 5\right)} + \left(\frac{\cos{\left(x \right)} \operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)} + 5} + \frac{2 \log{\left(\sin{\left(x \right)} + 5 \right)}}{\sqrt{1 - x^{2}}}\right)^{3} \operatorname{asin}^{3}{\left(x \right)} - 3 \left(\frac{\cos{\left(x \right)} \operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)} + 5} + \frac{2 \log{\left(\sin{\left(x \right)} + 5 \right)}}{\sqrt{1 - x^{2}}}\right) \left(- \frac{2 x \log{\left(\sin{\left(x \right)} + 5 \right)} \operatorname{asin}{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{\sin{\left(x \right)} \operatorname{asin}^{2}{\left(x \right)}}{\sin{\left(x \right)} + 5} + \frac{\cos^{2}{\left(x \right)} \operatorname{asin}^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} + 5\right)^{2}} + \frac{2 \log{\left(\sin{\left(x \right)} + 5 \right)}}{x^{2} - 1} - \frac{4 \cos{\left(x \right)} \operatorname{asin}{\left(x \right)}}{\sqrt{1 - x^{2}} \left(\sin{\left(x \right)} + 5\right)}\right) \operatorname{asin}{\left(x \right)} - \frac{\cos{\left(x \right)} \operatorname{asin}^{2}{\left(x \right)}}{\sin{\left(x \right)} + 5} + \frac{3 \sin{\left(x \right)} \cos{\left(x \right)} \operatorname{asin}^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} + 5\right)^{2}} + \frac{2 \cos^{3}{\left(x \right)} \operatorname{asin}^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} + 5\right)^{3}} - \frac{6 \cos{\left(x \right)}}{\left(x^{2} - 1\right) \left(\sin{\left(x \right)} + 5\right)} - \frac{6 \sin{\left(x \right)} \operatorname{asin}{\left(x \right)}}{\sqrt{1 - x^{2}} \left(\sin{\left(x \right)} + 5\right)} - \frac{6 \cos^{2}{\left(x \right)} \operatorname{asin}{\left(x \right)}}{\sqrt{1 - x^{2}} \left(\sin{\left(x \right)} + 5\right)^{2}} + \frac{2 \log{\left(\sin{\left(x \right)} + 5 \right)} \operatorname{asin}{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}}\right)$$

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Derivada de y=(5+sinx)^arcsinx^2

Gráfico:

Definida a trozos:

Solución