What is a derivative?
A derivative tells you the instantaneous slope at a specific point (for example, at \(x=2\)) — the slope of the tangent line to a curve.
Another way to say it: it’s the instantaneous rate of change.
Notation: \(f'(x)\), \(\dfrac{dy}{dx}\)
Start with slope between two points
If you have two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope is:
\[ m = \dfrac{y_2 - y_1}{x_2 - x_1} \]
That’s an average slope (a straight line between two points).
Example: \(f(x)=x^2\)
Pick two points on the curve: \((1,1)\) and \((2,4)\).
Average slope between them:
\[ m = \dfrac{4-1}{2-1} = 3 \]
\(y=x^2\)
Secant (two-point line)
Derivative = “points get closer”
The derivative is what the slope becomes as the second point moves closer and closer.
Formally, that idea is written with a limit:
\[ f'(x)=\lim_{h\to 0} \dfrac{f(x+h)-f(x)}{h} \]
For \(f(x)=x^2\), the derivative is:
\[ f'(x)=2x \]
Instantaneous slope at a point
At \(x=1\), the derivative is \(f'(1)=2\). That means the tangent line there has slope 2.
\(y=x^2\)
Tangent (instantaneous slope line)
Quick intuition: Average slope = “over a chunk.”
Derivative = “instantaneous (at a point).”
Derivative = “instantaneous (at a point).”
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