Best Error Function for Areas with Larger Slopes I have some nonlinear data (let's say x's and y's) that I would like to perform regression on, and I would like to focus on having the error lower on regions where the graph is more sloped, rather than the areas where the graph is flat, since that area will generally converge to a single value.
Currently, I am using mean squared error, which seems to give priority to the flat regions, so is there another error function I can use?
Alternatively, is there a statistically justified way to add points to the data at the regions where the slope is greater such that the error will be minimized on the sloped regions?

The first idea to come to mind is:
$$
\sum_{i=1}^n \bigg[\big(y_i-\hat y_i\big)^2
+
\bigg\vert\dfrac{
d\hat y
}{
dx
}\bigg\vert
\big(y_i-\hat y_i\big)^2
\bigg]
$$
This gives you the regular square loss but gives an additional penalty for missing when there is a large magnitude of slope.
This particular expression might not be the best for you, but it should start to give you some ideas. Variations could be using absolute loss instead of square loss (in either or both places the residuals are squared) and taking powers of the derivative.
If you have multiple features, you could use the gradient instead of the derivative.
