How did Newton find derivative of basic functions before formulating systematic Calculus? I think I read somewhere that Newton tried to find derivatives of basic functions like $x^2$ before formulating systematic calculus; how did/would he do it?
The earliest formulations of something that looks like calculus were not in terms of functions but of curves given by algebraic relations. So if we have
$$ y=x^2 $$
and $(y+p,x+q)$ is another point on the curve, we would also have
$$ (y+p) = (x+q)^2 $$
and if we multiply these equations out and subtract we get
$$ p = 2qx + q^2 $$

Now what happens in the earliest sources is that it is simply postulated, without any backing by theoretical definitions or limits that when $p$ and $q$ are small, we can remove their higher powers, so ignoring the $q^2$ term yields us
$$ p = 2qx \qquad\text{or, in other words,}\qquad p:q = 2x:1 $$
which can be used to draw a tangent.

The justification for this procedure was initially just that it worked in practice, but the unexplained ignoring of terms like $q^2$ drew quite a lot of contemporary criticism.

This was in the generation before Newton (chiefly Fermat and Descartes). They could produce painstaking geometric proofs in the Euclidean tradition (which was the gold standard for proofs in those days) that for each of the actual curves they considered what came out of this procedure was the right result -- but they didn't have the definitions and machinery to explain or prove rigorously why the procedure always works.

Newton brought in the new idea of a quantity $x$ that varied with time and its time rate of change $\dot x$, but he was still building on the "ignore higher powers of the increments" method. But he did it masterfully and was able to get more out of it than his predecessors did.
Surely any child with an interest in numbers will notice that in a quadratic sequence like 1, 4, 9, 16, 25, ..., the differences between successive terms form a linear sequence like 3, 5, 7, 9, ...? That doesn't need calculus; but someone with a sophisticated mathematical brain will start to recognise that there's a relationship between the function that generates the first sequence and the function that generates the second, and will develop such thinking to discover the principles of calculus.