really understand what the derivative is measuring, what it's telling you about the function. And so if we just look up derivative we see that okay so this is not a very good definition of derivative. This is a definition of what it means to be differentiable but if you remember from your calculus it is the limit as h goes to zero of f of x plus h minus f of x over h. So basically what it's saying is if you slightly bump up your at some point x that you're interested in or a and if you slightly bump up you know you slightly increase it by small number h how does the function respond with what sensitivity does it respond where is the slope at that point does the function go up or does it go down and by how much and that's the slope of that function the the slope of that response at that point and so we can basically evaluate the derivative here numerically by taking a very small h of course the definition would ask us to take h to zero we're just going to pick a very small h 0.001 and let's say we're interested in 0.3.0 so we can look at f of x of course as 20 and now f of x plus h so if we slightly nudge x in a positive direction how is the function going to respond and just looking at this do you expand do you expect f of x plus h to be slightly greater than 20 or do you expect it to be slightly lower than 20 and since this 3 is here and this is 20 if we slightly go positively the function will respond positively so you'd expect this to be slightly greater than 20 and now by how much is telling you the sort of the the strength of that slope right the the size of the slope so f of x plus h minus f of x this is how much the function responded in a positive direction and we have to normalize by the run so we have the rise over run to get the slope so this
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