Derivatives

  • Instantaneous rate of change: how fast is something going right now, not an average measurement.
  • Slope == average rate of change. Assuming time on the X-axis and distance on the Y-axis, with a straight line representing distance covered:
    $$ \text{Rate of change in distance} = \frac{\Delta y}{\Delta x} = \text{slope} $$
  • This works fine when you’re dealing with a straight line (avg), but how do you apply this when you have a constant stream of instantaneous measurements and the graph is a curve, not a straight line? Essentially, take infinitesmally small measurements:
    $$ \lim_{\Delta x\to0}\frac{\Delta y}{\Delta x} $$
  • Also known as the derivative, again, conceptualized as an infinitely small change in y over an infinitely small change in x:
    $$ \frac{dy}{dx} $$
  • A straight line has a constant rate of change; the slope doesn’t change. A curve has a rate of change that’s continously changing; the rate of change at a given point is the slope of a tangent to the line drawn at that point.
  • Notation variants (all these are equivalent):
    $$ \frac{dy}{dx} = f'(x_1) = \dot{y} = y' $$
  • distance over time → speed over time (first derivative) → acceleration over time (second derivative)
  • Derivative of a function; given an \(x\), choose a higher value \(x+h\). The slope of the tangent at \(x\) is:
    $$ f'(x) = \lim_{h\to0}\frac{f(x+h) + f(x)}{h} $$

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