If f(x) = 2sin(5x)arcsin(x), find f'(x).

just use the product rule:

2(5cos(5x)arcsin(x) + sin(5x)/√(1-x^2))

To find f'(x), we need to differentiate the given function f(x) = 2sin(5x)arcsin(x) using the chain rule and product rule.

Let's break down the function into its individual components:

1. The first component is 2sin(5x). This is a composite function where the outer function is sin(x) and the inner function is 5x. To differentiate sin(5x), we use the chain rule, which states that if g(x) = f(u(x)), then g'(x) = f'(u(x)) * u'(x). In this case, f(u) = sin(u) and u = 5x.

Using the chain rule, we differentiate sin(5x) as follows:
d/dx(sin(5x)) = cos(5x) * d/dx(5x) = 5cos(5x).

2. The second component is arcsin(x). To differentiate arcsin(x), we can use the formula d/dx(arcsin(x)) = 1 / sqrt(1 - x^2).

Now, let's apply the product rule to differentiate the entire function f(x) = 2sin(5x)arcsin(x):

f'(x) = (2sin(5x))' * arcsin(x) + 2sin(5x) * (arcsin(x))'.

Using the results from the individual differentiations above, we have:
f'(x) = (5cos(5x)) * arcsin(x) + 2sin(5x) * (1 / sqrt(1 - x^2)).

Therefore, the derivative of f(x) = 2sin(5x)arcsin(x) is:
f'(x) = 5cos(5x) * arcsin(x) + 2sin(5x) / sqrt(1 - x^2).