Limits: lim
limₓ→ₐ f(x) describes the value approached as x approaches a; f(a) need not be defined. For example, limₓ→₀ sin(x)/x = 1 when angles are measured in radians.
Derivatives: d, ′ and ∂
f′(x) and df/dx commonly denote a one-variable derivative. If f(x) = x², then f′(x) = 2x. For f(x, y) = x² + y², the partial derivative ∂f/∂x = 2x holds y constant. Ordinary and partial derivatives are not interchangeable decorations.
Integrals: ∫
The indefinite integral ∫2x dx = x² + C is a family of antiderivatives and needs the constant C. The definite integral ∫₀¹ x² dx = 1/3 is a number. The integrand, bounds and integration variable determine the meaning.
Sums and gradients: ∑ and ∇
∑ᵢ₌₁ⁿ i = n(n + 1)/2 is a finite sum. For f(x, y) = x² + y², the gradient is ∇f = (2x, 2y). ∇·F denotes divergence, while ∇×F denotes curl in three dimensions.
LaTeX reference
\lim_{x \to 0}, \frac{df}{dx}, \frac{\partial f}{\partial x}, \int_0^1 x^2\,dx, \sum_{i=1}^{n} i, and \nabla f. Use math mode and adapt bounds and variables to the problem.
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