Calculus

This table describes how to type calculus expressions, along with usage examples (including the typeset equivalent.) If the expression you need is not listed, please define your own and explain the meaning of your notation.

Expression	Description	Examples
$$\{a_n\}=\{a_1,a_2,\cdots a_n,\cdots\}$$	An infinite sequence.	$$\{3+(-1)^n\}$$ $$=\{2,4,2,4,\cdots\}$$ { 3 + (-1)^n} = {2, 4, 2, 4, ...}
$$\sum_{n=1}^{\infty}=a_1+a_2+\cdots+a_n=\cdots$$	An infinite series.	$$\sum_{n=1}^{\infty}\frac{1}{2^n}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$$ sum[n = 1, oo] 1/(2^n) = (1/2) + (1/4) + (1/8) + ...
$$\lim_{x \to c} f(x)$$	Limit of $f(x)$ as $x$ approaches $c$ .	$$\lim_{x \to 0} x^2=0$$ lim x->0 (x^2) = 0
$$dx \quad dy$$	Differentials. The differential of x. The differential of y.	dx dy
$$y'$$	y prime. The first derivative of y.	$$y=x^2$$ $$y'=2x$$ y = x^2 y' = 2x
$$y''$$	The second derivative of y.	$$y=x^2$$ $$y''=2$$ y = x^2 y'' = 2
$$y'''$$	The third derivative of y.	$$y=x^2$$ $$y'''=0$$ y = x^2 y''' = 0
$$f'(x)$$	f prime of x, the first derivative of f.	$$f(x)=x^2$$ $$f'(x)=2x$$ f(x) = x^2 f'(x) = 2x
$$f''(x)$$	The second derivative of f.	$$f(x)=x^2$$ $$f''(x)=2$$ f(x) = x^2 f''(x) = 2
$$f^{(n)}(x)$$	Higher order derivatives. The $\mathrm{n^{th}}$ derivative of f.	$$f(x)=x^3$$ $$f^{(4)}(x)=0$$ f(x) = x^3 f^(4)(x) = 0
$$\frac{dy}{dx}$$ $$\frac{d}{dx}[x^2]$$	Another commonly used notation for derivatives. The first is read ``the derivative of y with respect to x.''	$$y=x^2$$ $$\frac{dy}{dx}=2x$$ $$\frac{d}{dx}[x^2]=2x$$ y = x^2 dy/dx = 2x d/dx[x^2] = 2x
$$\frac{d^{2}y}{dx^{2}}$$	The second derivative of y with respect to x.	$$y=x^2$$ $$\frac{d^{2}y}{dx^{2}}=2$$ y = x^2 (d^2y)/(dx^2) = 2
$$\frac{d^{n}y}{dx^{n}}$$	Higher order derivatives. The $\mathrm{n^{th}}$ derivative of y with respect to x.	$$y=x^3$$ $$\frac{d^{4}y}{dx^{4}}=0$$ y = x^3 (d^4y)/(dx^4) = 0
$$\frac{\partial z}{\partial x}$$ $$\frac{\partial}{\partial x}[f(x,y)]$$	Partial derivatives. The first is read ``the partial of z with respect to x.''	$$z=x^2+xy$$ $$\frac{\partial z}{\partial x}=2x+y$$ $$\frac{\partial}{\partial x}\Big[x^2+xy\Big]=2x+y$$ z = x^2 + xy pz/px = 2x + y p/px[x^2 + xy] = 2x + y
$$F$$	An antiderivative of f.	$$f(x)=3x^2$$ $$F(x)=x^3$$ f(x) = 3x^2 F(x) = x^3
$$\int f(x)dx$$	Indefinite integration.	$$\int x^2 dx=\frac{x^3}{3}+C$$ int(x^2) dx = (x^3) /3 + C
$$\int_{a}^{b}f(x)dx$$	Definite integration.	$$\int_{0}^{2} \left.x^2 dx =\frac{x^3}{3}\right]_{0}^{2}$$ $$=\frac{8}{3}$$ int[0,2](x^2) dx = (x^3) /3 ]0,2 = 8/3
$$\int_{a}^{b}\!\!\! \int_{c}^{d}f(x,y)\:dy dx$$	Multiple integration. A double integral, in this case.	$$\int_{0}^{1}\!\!\! \int_{0}^{2}(x+y)\:dy dx$$ $$\int_{0}^{\frac{\pi}{2}}\!\!\! \int_{0}^{2\cos\theta}r\:dr d\theta$$ int[0,1]int[0,2] (x+y) dydx int[0,(pi/2)]int[0,2cos(theta)]r drd(theta)