Define the function f(x)=2x/(1−x^2). Find the number of distinct real solutions of the equation f^(5)(x)=x.

This is ambiguous notation. Do you mean

5th power of f(x)
5th derivative of f(x)
f(f(f(f(f(x)))))

Hard to say from what is here.

i mean f^(5)(x)=f(f(f(f(f(x))))).

To find the number of distinct real solutions of the equation f^(5)(x) = x, we need to determine the roots of the equation by solving it algebraically.

Start by finding the fifth derivative of the function f(x) = 2x/(1 - x^2).

First, let's find the first derivative of f(x):
f'(x) = d/dx (2x/(1 - x^2))

To differentiate f(x), we can use the quotient rule:
f'(x) = (2(1 - x^2) - 2x(-2x))/(1 - x^2)^2
= (2 - 2x^2 + 4x^2)/(1 - x^2)^2
= (2 + 2x^2)/(1 - x^2)^2

Now, let's find the second derivative of f(x):
f''(x) = d/dx (f'(x))
= d/dx ((2 + 2x^2)/(1 - x^2)^2)

Using the quotient rule again, we have:
f''(x) = [(2 - 8x^2 + 4x(2x))/((1 - x^2)^2)^2
= (2 - 8x^2 + 8x^2)/(1 - x^2)^3
= (2 - 8x^2)/(1 - x^2)^3

Next, let's find the third derivative of f(x):
f'''(x) = d/dx (f''(x))
= d/dx ((2 - 8x^2)/(1 - x^2)^3)

Applying the quotient rule again, we have:
f'''(x) = [(0 - (-16x(1 - x^2)))/(1 - x^2)^6
= (16x(1 - x^2))/(1 - x^2)^6
= 16x/(1 - x^2)^5

Finding the fourth derivative of f(x):
f^4(x) = d/dx (f'''(x))
= d/dx (16x/(1 - x^2)^5)

Using the quotient rule once more, we have:
f^4(x) = [(16 - 4x^2)/(1 - x^2)^5

Finally, let's find the fifth derivative of f(x):
f^5(x) = d/dx (f4(x))
= d/dx (16 - 4x^2)/(1 - x^2)^5)

Using the quotient rule again, we have:
f^5(x) = [(0 - 8x(1 - x^2))/(1 - x^2)^10
= (-8x + 8x^3)/(1 - x^2)^10

Now that we have the fifth derivative, we can create an equation to find the roots of f^5(x) = x:
(-8x + 8x^3)/(1 - x^2)^10 = x

Simplifying the equation, we get:
-8x + 8x^3 = (1 - x^2)^10 * x
-8x + 8x^3 = (1 - x^2)^10x

To find the number of distinct real solutions, we need to solve this equation by factoring, using the quadratic formula, or another appropriate method. However, factoring or finding an analytical solution for this equation is not straightforward. Therefore, to find the exact number of distinct real solutions, numerical methods such as graphing or using a calculator or software can be used to approximate the solutions.