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.
Open →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 →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 →Key Rules
The core differentiation and integration 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
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Common Questions
What is the Fundamental Theorem of Calculus?
The Fundamental Theorem of Calculus links differentiation and integration. Part 1 states that if F is an antiderivative of f, then the definite integral from a to b of f(x)dx equals F(b) - F(a). Part 2 states that the derivative of an accumulation function is the original integrand.
When should you use the chain rule?
Use the chain rule whenever you differentiate a composite function f(g(x)). Multiply the derivative of the outer function evaluated at the inner function by the derivative of the inner function: d/dx[f(g(x))] = f'(g(x)) * g'(x).