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more specifically, the derivative of a function allows us to calculate the slope (rate of change) of non-linear functions
for example, for the function f(x) = x^2, the derivative is 2x. when x = 1, the derivative tells us that the "slope" (rate of change) is 2, and when x = 2, the slope is 4 this can be represented by drawing a tangent line at the specified x value
Do you remember back to algebra when you would find the slope of a line like this?|dw:1443581354508:dw|This is a secant line, a line passing through two distinct points. Another way to think of this is that you're calculating `average rate of change`. In calculus we're interested in finding `instantaneous rate of change`, how much is something changing at an exact moment? That's what the derivative does for us :)
using derivatives (generation of linear functions) can be used to model nonlinear functions in the real world that are too complex to solve ; a unique way of applying derivatives
The definition of the derivative tells us... that we're letting the space between those two points get smaller and smaller,|dw:1443581597040:dw|
taking this idea to the limit, meaning we let our space between the points get really really really small, until the two points are immeasurably close together, is what gives us our "tangent" line.|dw:1443581806796:dw|
she gone, she was my picture and was like i gotz it , pz
And this is what the derivative does for us, it gives us the slope value at particular point, not over some interval, but at a specific moment in time! It has incredible implications. Imagine you're driving on the highway, what's less important is what average speed you went during your trip. more important is how fast you were going at any particular moment, because you certainly don't want to be caught speeding! :O So there is an instance where instantaneous rate of change is awesome sauce.
haha ya i see that XD lameeee