Find probability density function of $Y=X^2$ for uniform $X.$ Suppose that a random variable $X$ has a uniform distribution on the interval $[0,1]$. Determine the p.d.f of $X^2$
Following this equation:
$$f(x=y)\frac{dx}{dy}$$
I then solve for $Pr(Y\le y)=Pr(X^2 \le y)=Pr(X\le y^{\frac{1}{2}})$ so that $x=y^{\frac{1}{2}}$
Then taking the derivative of $\sqrt{y}$ I get $\frac{1}{2}y^{-\frac{1}{2}}$
I should get $$y^{\frac{1}{2}}\frac{1}{2}y^{-\frac{1}{2}}=\frac{1}{2}$$
I have taken this from Degroot's Probability and Statistics (third edition- page 164 exercise7.) However the solution is actually
$$1 \cdot \frac{1}{2}y^{-\frac{1}{2}} $$
What is the explanation behind this?
Added comments:
$G(y) = Pr(Y \le y) = Pr[r(X) \le y]=Pr[X \le s(y)] = F[s(y)]$
If it is also assumed that s is a differentiable function over the interval $(\alpha, \beta)$, then the distribution of $Y$ will be continuous and the value of its p.d.f $g(y)$ for $\alpha < y < \beta$ will be:
$g(y) = \frac{dG(y)}{dy}=\frac{dF[s(y)]}{dy}=f[s(y)]\frac{ds(y)}{dy}$
In other words, the pdf $g(y)$ of $Y$ can be obtained directly from the pdf $f(x)$ by replacing $x$ but its expression in terms of $y$ and multiplying the result by $dx/dy$

If $F_Y(y) = y^{1/2},$ for $0 < y < 1,$ then
$F_Y^\prime (y) = f_Y(y) = \frac 12 y^{-1/2},$ which is the density
function of the beta distribution $\mathsf{Beta}(1/2, 1),$
with mean $E(Y) = 1/3.$
Illustration of this 'CDF method' by simulation:
set.seed(812)
x = runif(10^5);  y = x^2
summary(y)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
0.00000 0.06275 0.24826 0.33192 0.55900 0.99999 

hdr = "Simulated Dist'n of Y with Density of BETA(.5,1)"
hist(y, prob=T, br=30, col="skyblue2", main=hdr)
 curve(dbeta(x, .5, 1), add=T, lwd = 3, col="orange")


Addendum: In the histograms below, images and objects of the transformation are represented by bars of the same color; all bars have (very nearly) the same area. [In the 'PDF method', the shape of the beta histogram arises from $dx/dy.]$

set.seed(2021)
x = runif(10^5);  y = x^2
cutp = seq(0,1,by=.1)
par(mfrow=c(1,2))
 hist(x, prob=T, br=cutp, ylim=c(0,10), col=rainbow(11), main="UNIF(0,1)")
 hist(y, prob=T, br=cutp^2, col=rainbow(11), main="BETA(.5,1)")
par(mfrow=c(1,1))

