Hi forum, any help appreciated!

Notationis the cumulative distribution of x

is the probability density of x

Problem

Let X be a random variable with a uniform distribution on (-1,1)

Let [a is a constant between 0 and 1]

Find the pdf of Y.

My attempt (ive highlighted in red where i assume my mistake is)i will use these results later:

Now try to find and then differentiate to get

(apply inverse of g to both sides of inequality)

(simplify)

(the RHS is by definition the CDF of x)

(we know the functional form of the CDF of x is 0.5x+0.5, substitute)

Now we can differentiate (this is where i a assume my mistake is)

use the rule that the derivative of is the reciprocal of the derivative of g.

now, substitute the actual

ive done some monte carlo simulations and im pretty sure my CDF(y) is correct but my PDF(y) isn't .... can anyone help?