Why can a derivative be non-linear?

Edited

OK, so I'll try to make this post more self contained.

To find an equation of the tangent line to any function $y = f(x)$ at the point $A(x_0, y_0)$, iff $f(x)$ is differentiable at that point, one needs to calculate its algebraic form as $$ y = y_0 + f'(x_0)(x - x_0) $$ so, as you can see, even though you calculate $f'(x)$ to find tangent line slope, it's evaluated at $x_0$, which makes it a number, but that number will change when you move from point to point.

Now, let's consider that example $$ y = x^3 \implies y' = 3x^2 \implies y = x_0^3 + 3x_0^2(x - x_0) = 3x_0^2 x - 2x_0^3 $$ Simple animation of the tangent line when point it was calculated at is moving provided below.

The slope of a single tangent line is a constant: it's the $m$ in the equation $y=mx+b$ for that line. So if the derivative of $f(x)$ is $2x$ (for example) then you already have a different slope at each point of $y=f(x)$, and therefore an entirely different tangent line at each point.

The derivative at a precise point $x$ is the slope of the tangent line at this point. But the derivative is a function so the slope is moving while $x$ is moving. Actually the derivative can be viewed as the variation of the slope of the tangent line.

For exemple, the slope of the tangent line of the function $x \to x^3$ increases faster than the slope of $x \to x^2$. In the first case quadratically and in the second linearly.