8arcsin(x^(2)),
find f '(x).
Find f '(0.3).
To find f'(x) for the function f(x) = 8arcsin(x^2), we need to apply the chain rule. The chain rule states that if we have a composite function, f(g(x)), then its derivative is given by f'(g(x)) * g'(x).
Let's break down the function f(x) = 8arcsin(x^2) into the composition of two functions:
1. The outer function f(x) = 8arcsin(x).
2. The inner function g(x) = x^2.
Taking the derivative of the inner function g(x) = x^2:
g'(x) = 2x.
Now, let's find the derivative of the outer function f(x) = 8arcsin(x):
f'(x) = 8 * (d(arcsin(x))/dx).
To find the derivative of arcsin(x), we can use the inverse trigonometric identity:
d(arcsin(x))/dx = 1 / sqrt(1 - x^2).
Therefore,
f'(x) = 8 * (1 / sqrt(1 - x^2)) * 2x.
Simplifying further, we get:
f'(x) = 16x / sqrt(1 - x^2).
To find f'(0.3), substitute x = 0.3 into the equation we derived for f'(x):
f'(0.3) = 16 * 0.3 / sqrt(1 - (0.3)^2).
Now calculate the value of f'(0.3) using a calculator or by simplifying the expression further.