How Derivatives Work
It Starts With Slope
How steep is the curve at one exact point?
The slope of the tangent line tells you how fast f(x) is changing at exactly that point. The derivative is just this: the instantaneous rate of change.
Step 1 of 6: It Starts With Slope
How steep is the curve at one exact point?
The slope of the tangent line tells you how fast f(x) is changing at exactly that point. The derivative is just this: the instantaneous rate of change.
Step 2 of 6: Two Points, One Line
Drag the second point closer — watch what happens to the slope
The secant line connects two points on the curve. Its slope = [f(a+h) - f(a)] / h. As you drag h smaller, the secant slope gets closer to the true derivative.
Step 3 of 6: Shrinking Toward Zero
The secant line becomes a tangent as h vanishes
The secant cuts through the curve at two points. As h shrinks, the second point slides toward the first.
Step 4 of 6: The Algebra Behind It
Step through the limit computation for f(x) = x²
The derivative is defined as this limit as h approaches 0.
Step 5 of 6: The Derivative Function
Every x has its own slope — that gives us a whole new function
The curve is climbing here — f'(x) is positive. The steeper the curve, the higher the derivative.
Step 6 of 6: Why It Matters
Derivatives power everything from physics to machine learning
The derivative is one of the most powerful ideas in mathematics. It turns the question "how fast is this changing?" into a precise, computable answer. Everything from rocket trajectories to training AI models depends on it.