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y=arcsin^8(1-x^1/3)

Derivada de y=arcsin^8(1-x^1/3)

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Solución

Ha introducido [src]
    8/    3 ___\
asin \1 - \/ x /
$$\operatorname{asin}^{8}{\left(1 - \sqrt[3]{x} \right)}$$
asin(1 - x^(1/3))^8
Gráfica
Primera derivada [src]
            7/    3 ___\     
     -8*asin \1 - \/ x /     
-----------------------------
           __________________
          /                2 
   2/3   /      /    3 ___\  
3*x   *\/   1 - \1 - \/ x /  
$$- \frac{8 \operatorname{asin}^{7}{\left(1 - \sqrt[3]{x} \right)}}{3 x^{\frac{2}{3}} \sqrt{1 - \left(1 - \sqrt[3]{x}\right)^{2}}}$$
Segunda derivada [src]
                    /                       /     3 ___\     /     3 ___\              /     3 ___\     \
      6/     3 ___\ |          7            \-1 + \/ x /*asin\-1 + \/ x /        2*asin\-1 + \/ x /     |
8*asin \-1 + \/ x /*|- ------------------ + ----------------------------- - ----------------------------|
                    |                   2                         3/2                 __________________|
                    |       /     3 ___\        /               2\                   /                2 |
                    |  -1 + \-1 + \/ x /        |    /    3 ___\ |          3 ___   /      /    3 ___\  |
                    \                           \1 - \1 - \/ x / /          \/ x *\/   1 - \1 - \/ x /  /
---------------------------------------------------------------------------------------------------------
                                                     4/3                                                 
                                                  9*x                                                    
$$\frac{8 \left(- \frac{7}{\left(\sqrt[3]{x} - 1\right)^{2} - 1} + \frac{\left(\sqrt[3]{x} - 1\right) \operatorname{asin}{\left(\sqrt[3]{x} - 1 \right)}}{\left(1 - \left(1 - \sqrt[3]{x}\right)^{2}\right)^{\frac{3}{2}}} - \frac{2 \operatorname{asin}{\left(\sqrt[3]{x} - 1 \right)}}{\sqrt[3]{x} \sqrt{1 - \left(1 - \sqrt[3]{x}\right)^{2}}}\right) \operatorname{asin}^{6}{\left(\sqrt[3]{x} - 1 \right)}}{9 x^{\frac{4}{3}}}$$
Tercera derivada [src]
                    /                                                                                                                                                                 2                                                     \
                    |                                  2/     3 ___\                  2/     3 ___\                /     3 ___\            2/     3 ___\ /     3 ___\     /     3 ___\      2/     3 ___\      /     3 ___\     /     3 ___\|
      5/     3 ___\ |           42                 asin \-1 + \/ x /           10*asin \-1 + \/ x /         42*asin\-1 + \/ x /      6*asin \-1 + \/ x /*\-1 + \/ x /   3*\-1 + \/ x / *asin \-1 + \/ x /   21*\-1 + \/ x /*asin\-1 + \/ x /|
8*asin \-1 + \/ x /*|------------------------ + ------------------------ + --------------------------- + ------------------------- - -------------------------------- + --------------------------------- + --------------------------------|
                    |                     3/2                        3/2            __________________        /                 2\                             3/2                                5/2                                  2    |
                    |   /               2\         /               2\              /                2     7/3 |     /     3 ___\ |           /               2\                 /               2\                 /                 2\     |
                    | 2 |    /    3 ___\ |       2 |    /    3 ___\ |       8/3   /      /    3 ___\     x   *\-1 + \-1 + \/ x / /       7/3 |    /    3 ___\ |               2 |    /    3 ___\ |               2 |     /     3 ___\ |     |
                    \x *\1 - \1 - \/ x / /      x *\1 - \1 - \/ x / /      x   *\/   1 - \1 - \/ x /                                    x   *\1 - \1 - \/ x / /              x *\1 - \1 - \/ x / /              x *\-1 + \-1 + \/ x / /     /
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                                                                                                                      27                                                                                                                     
$$\frac{8 \left(\frac{21 \left(\sqrt[3]{x} - 1\right) \operatorname{asin}{\left(\sqrt[3]{x} - 1 \right)}}{x^{2} \left(\left(\sqrt[3]{x} - 1\right)^{2} - 1\right)^{2}} + \frac{\operatorname{asin}^{2}{\left(\sqrt[3]{x} - 1 \right)}}{x^{2} \left(1 - \left(1 - \sqrt[3]{x}\right)^{2}\right)^{\frac{3}{2}}} + \frac{42}{x^{2} \left(1 - \left(1 - \sqrt[3]{x}\right)^{2}\right)^{\frac{3}{2}}} + \frac{3 \left(\sqrt[3]{x} - 1\right)^{2} \operatorname{asin}^{2}{\left(\sqrt[3]{x} - 1 \right)}}{x^{2} \left(1 - \left(1 - \sqrt[3]{x}\right)^{2}\right)^{\frac{5}{2}}} + \frac{42 \operatorname{asin}{\left(\sqrt[3]{x} - 1 \right)}}{x^{\frac{7}{3}} \left(\left(\sqrt[3]{x} - 1\right)^{2} - 1\right)} - \frac{6 \left(\sqrt[3]{x} - 1\right) \operatorname{asin}^{2}{\left(\sqrt[3]{x} - 1 \right)}}{x^{\frac{7}{3}} \left(1 - \left(1 - \sqrt[3]{x}\right)^{2}\right)^{\frac{3}{2}}} + \frac{10 \operatorname{asin}^{2}{\left(\sqrt[3]{x} - 1 \right)}}{x^{\frac{8}{3}} \sqrt{1 - \left(1 - \sqrt[3]{x}\right)^{2}}}\right) \operatorname{asin}^{5}{\left(\sqrt[3]{x} - 1 \right)}}{27}$$
Gráfico
Derivada de y=arcsin^8(1-x^1/3)