Drag point $Q$ along the curve to change interval width $h$
1. Average Rate of Change
To find the instantaneous rate of change at a point $P$, we start by looking at an average rate of change over an interval. By picking a nearby point $Q$ at a distance $h$ away, we can draw a secant line through both points.
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Drag the $h$ slider to see the secant line morph into the tangent line
2. Shrinking the Interval ($h \to 0$)
Now let's apply this to a concrete function: $f(x) = x^2$. We want to find the exact slope at any point $x$. Let's set up the algebra and observe what happens as we make the interval size $h$ approach $0$.
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Drag point $x$ to see the tangent slope match the derivative function $f'(x) = 2x$
3. The Formal Limit
By taking the limit as $h$ goes to $0$, we eliminate the interval entirely. The secant slope becomes the tangent slope, giving us the instantaneous rate of change: the derivative.
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