# Derivatives

Wiki Index

- 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:$$ f'(x) = \lim_{h\to0}\frac{f(x+h) + f(x)}{h} $$