# 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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