Calculus Topic
Derivatives and Integrals
The two fundamental operations of calculus. Derivatives measure instantaneous rates of change. Integrals accumulate quantities over an interval. Together they form the foundation of modern science and engineering.
Learning Path
Derivatives Poster
Visual reference for differentiation rules including power, product, quotient, and chain rules. Covers common derivatives and their geometric interpretation as slopes.
View Poster →Derivative and Integral Visualizer
Plot f(x), f'(x), and F(x) together on one graph. Drag a point to see the tangent line slope and accumulated area update in real time with step-by-step differentiation.
Open Tool →Derivatives and Tangent Lines Lab
Visualize derivatives as slopes of tangent lines. Animate secant lines approaching tangents as h approaches 0, explore derivative rules step by step, and compute slopes on implicit curves.
Open Lab →Key Rules
- Power rule: d/dx[x^n] = nx^(n-1)
- Product rule: (uv)' = u'v + uv'
- Chain rule: d/dx[f(g(x))] = f'(g(x))g'(x)
- FTC: d/dx[integral from a to x of f(t)dt] = f(x)
- Integral of x^n: x^(n+1)/(n+1) + C
More Resources
Integrals Poster
Visual reference for integration techniques including substitution, integration by parts, and common antiderivatives with geometric interpretations.
View Poster →FTC and Net Change Lab
Explore the Fundamental Theorem of Calculus by building accumulation functions, evaluating definite integrals with antiderivatives, and computing displacement versus total distance.
Open Lab →Riemann Sum Explorer
Approximate definite integrals with left, right, midpoint, and trapezoidal methods. Use the slider to add rectangles and watch the approximation converge to the exact value.
Open Tool →