Let f(x) = y = arcsin^3(5x) = (arcsin(5x))^3.
We need to use the chain rule here.
Recall that (dy)/(dx) = (du)/(dx) * (dy)/(du).
First let u = arcsin(5x) and y=u^3.
To solve for (dy)/(du) where y = u^3, we can use the power rule:
(dy)/(du) = 3u^2
To solve for (du)/(dx) where u = arcsin(5x), we can use the inverse sin derivative rule, combined with the chain rule again.
Let u' = 5x, so u = arcsin(5x) = arcsin(u').
By the inverse sin derivative rule:
(du)/(du') = 1/(sqrt(1-u'^2))
By the power rule:
(du')/(dx) = 5
Then multiplying together and substituting in for u', we have:
(du)/(dx) = (du)/(du') * (du')/(dx)
(du)/(dx) = 5/(sqrt(1-(5x)^2))
(du)/(dx) = 5/(sqrt(1-25x^2))
Finally, we have:
(dy)/(dx) = (du)/(dx) * (dy)/(du)
(dy)/(dx) = 5/(sqrt(1-25x^2)) * 3u^2
Plugging back in u and simplifying:
(dy)/(dx) = 5/(sqrt(1-25x^2)) * 3arcsin(5x)^2
(dy)/(dx) = (15arcsin(5x)^2)/(sqrt(1-25x^2))
