Derivatives
The Derivative
The single most important idea in calculus: how fast something is changing at a particular instant.
Definition
The derivative of a function $f$ at a point $a$ is defined as the limit of the average rate of change as the interval shrinks to zero:
Provided this limit exists, we say $f$ is differentiable at $a$. The value $f'(a)$ represents the instantaneous rate of change of $f$ at $a$, and geometrically equals the slope of the tangent line to the curve $y = f(x)$ at the point $(a, f(a))$.
Notation
Several equivalent notations exist:
Each has its place. Lagrange's $f'(x)$ is compact. Leibniz's $\frac{dy}{dx}$ makes the chain rule transparent. Newton's $\dot{y}$ persists in physics for time derivatives.
Differentiability and continuity
If $f$ is differentiable at $a$, then $f$ is continuous at $a$. The converse is false: continuous functions can fail to be differentiable. The classic counterexample is $f(x) = |x|$ at $x = 0$ — continuous everywhere, but the left and right limits of the difference quotient disagree, so the derivative does not exist there.
Worked example
Find $f'(2)$ for $f(x) = x^2$.
So the slope of $y = x^2$ at $x = 2$ is exactly $4$.
The power rule (preview)
Computing every derivative from the limit definition would be tedious. The power rule states that for $f(x) = x^n$, we have $f'(x) = nx^{n-1}$. This single rule, combined with linearity, handles all polynomials. We'll prove and extend it in the next topic.
The simple version
The derivative answers one question: how fast is something changing right now?
Think about driving. Your speedometer doesn't show "average speed over the trip" — it shows your speed at this exact second. That's a derivative. Position is the function; speed is its derivative.
Slope of a curve
For a straight line, slope is easy: rise over run. But what's the "slope" of a curve, which keeps changing direction? You zoom in.
If you zoom in close enough on any smooth curve, it starts to look like a straight line. The derivative is the slope of that "zoomed-in straight line" at a single point. Mathematicians call this the tangent line.
The formula, demystified
The official definition looks scary:
But it's just "rise over run" with a twist:
• Rise: $f(a+h) - f(a)$ — how much $y$ changed
• Run: $h$ — how much $x$ changed
• The trick: shrink $h$ to zero — the slope between two points becomes the slope at one point.
Why "$x$ to the something" gets simpler
Computing every derivative from that limit would take forever. Luckily there's a shortcut for power functions:
So $f(x) = x^2$ gives $f'(x) = 2x$. At $x = 2$, that's $4$ — the slope. No limit needed.
The whole concept in 8 words
The derivative tells you the slope at one point.
Derivative mastered!
You crushed it. Slope at every point, on lock.