P1
Derivative of a polynomial
Find f′(x) for f(x) = 3x² + 2x − 5.
Apply the power rule term by term: d/dx[xⁿ] = n·xⁿ⁻¹. Constants vanish.
1 · f′(x) = d/dx[3x²] + d/dx[2x] − d/dx[5]
2 · = 3·2x + 2·1 − 0
3 · = 6x + 2
→ f′(x) = 6x + 2
interactive calculus · made visible
See the math
before you trust it.
Drag, slide, and zoom through derivatives, integrals and limits — with the rigor of a textbook and the feel of a lab bench.
01 — Derivatives
Drag the point. Watch the slope.
The tangent line is the derivative made visible. Grab the point on f(x) = x²/4 − 2 and feel the slope change — rise over run, at an instant.
02 — Integrals
Riemann sums: approximate, then converge.
The integral is the limit of a sum. Slide the partition to refine — more rectangles, closer truth. The area under f(x) = 4 − x²/4 from −2 to 2.
03 — Limits
The ε–δ game.
Pick an ε — a tolerance around the limit L. The δ snaps to the largest interval that guarantees f(x) stays inside. That's the formal definition, played by hand.
04 — Practice
Exam-ready problems.
Four problems, each with a hint and a step-through solution. Press → to advance the proof one line at a time.
P2
Area under a curve
Compute ∫₀² (2x) dx.
An antiderivative of 2x is x². Evaluate at the bounds.
1 · ∫₀² 2x dx = [x²]₀²
2 · = (2)² − (0)²
3 · = 4
→ the area is 4
P3
Limit by factoring
Evaluate limx→3 (x² − 9)/(x − 3).
Factor the numerator as a difference of squares — the (x − 3) cancels.
1 · (x² − 9)/(x − 3) = (x − 3)(x + 3)/(x − 3)
2 · = x + 3, for x ≠ 3
3 · lim = 3 + 3
→ the limit is 6
P4
Second derivative
Find f″(x) for f(x) = sin(x).
Differentiate twice; recall the derivative cycle of sin and cos.
1 · f′(x) = cos(x)
2 · f″(x) = d/dx[cos(x)]
3 · = −sin(x)
→ f″(x) = −sin(x)