Lets say I have a polynomial $ax^4 +bx^3 +cx^2 +dx +e$ and the uncertainties on each coefficient. Now I need to calculate the tangent at some points as well as some areas under this curve. How would I find the uncertainties on those results?
My initial idea was to simply differentiate or integrate it and then use the partial derivatives method on the obtained expression(ignoring the error on "x" itself). ex. $slope = 4ax^3 + 3bx^2 +2cx +d$
$$\sigma_{slope}^2 = (4x^3)^2*\sigma_a^2 + (3x^2)^2*\sigma_b^2 + (2x)^2*\sigma_c^2 + \sigma_d^2$$
But higher x will make this error much higher which doesn't seem logical to me. I couldn't find the answer online and my book doesn't cover it.
Whats the right way to propagate uncertainties in such cases?