A function $f:(-1,1)\to \mathbb R$ which is differentiable and uniformly continuous but $f'$ is unbounded on the interval.

Question

I am finding an example of a function $f:(-1,1)\to \mathbb R$ which is differentiable and uniformly continuous but $f'$ is unbounded on the interval. I constructed the function $f(x)=\frac{4x^3}{1+x^4} .\sin(\frac{1}{x^{10}})$ if $x\neq 0$ and $f(x)=0$ if $x=0$ defined on the interval $(-1,1)$.I cannot find any flaw in my answer and I think it is a valid one.Can someone tell me if my answer is alright?