we have to normalize by the run so we have the rise over run to get the slope so this of course is just a numerical approximation of the slope because we have to make h very very small to converge to the exact amount now if i'm doing too many zeros at some point i'm going to i'm going to get an incorrect answer because we're using floating point arithmetic and the representations of all these numbers in computer memory is finite and at some point we get into trouble so we can converge towards the right answer with this approach but basically at 3 the slope is 14 and you can see that by taking 3x squared minus 4x plus 5 and differentiating it in our head so 3x squared would be 6x minus 4 and then we plug in x equals 3 so that's 18 minus 4 is 14 so this is correct so that's at 3 now how about the slope at say negative 3 would you expect what would you expect for the slope now telling the exact value is really hard but what is the sign of that slope so at negative 3 if we slightly go in the positive direction at x the function would actually go down and so that tells you that the slope would be negative so we'll get a slight number below below 20 and so if we take the slope we expect something negative negative 22 okay and at some point here of course the slope would be zero now for this specific function i looked it up previously and it's at point uh 2 over 3 so at roughly 2 over 3 that's somewhere here this this derivative would be zero so basically at that precise point yeah at that precise point if we nudge in a positive direction the function doesn't respond this stays the same almost and so that's why the slope is zero okay now let's look at a bit more complex case so we're going to start you know complexifying a bit so now we have a function here with output variable
rtdocs_stable/api.python.langchain.com/en/stable/modules/indexes/document_loaders/examples/audio.html