Calculus

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Calculus

61. Derivative of a Function: $f^{\mathrm{\prime }}(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

62. Power Rule: $\frac{d}{dx}{x}^n=n{x}^{n-1}$

63. Product Rule: $\frac{d}{dx}(uv)=u^{\mathrm{\prime }}v+u{v}^{\mathrm{\prime }}$

64. Quotient Rule: $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u^{\mathrm{\prime }}v-u{v}^{\mathrm{\prime }}}{v^2}$

65. Chain Rule: $\frac{d}{dx}f(g(x))=f^{\mathrm{\prime }}(g(x))g^{\mathrm{\prime }}(x)$

66. Integral of a Function: $\int f(x)dx$

67. Fundamental Theorem of Calculus: ${\int }_a^{b}f(x)dx=F(b)-F(a)$

68. Integration by Parts: $\int udv=uv-\int vdu$

69. Trapezoidal Rule: ${\int }_a^{b}f(x)dx\approx \frac{h}{2}(f(x_0)+2f(x_1)+\cdots +2f(x_{n-1})+f(x_n))$

70. Simpson's Rule: ${\int }_a^{b}f(x)dx\approx \frac{h}{3}(f(x_0)+4f(x_1)+2f(x_2)+\cdots +4f(x_{n-1})+f(x_n))$

71. Limit Definition: ${\lim }_{x\to a}f(x)=L$

72. L'Hôpital's Rule: ${\lim }_{x\to a}\frac{f(x)}{g(x)}={\lim }_{x\to a}\frac{f^{\mathrm{\prime }}(x)}{g^{\mathrm{\prime }}(x)}$

73. Taylor Series: $f(x)={\sum }_{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(a)}{n!}(x-a{)}^n$

74. Maclaurin Series: $f(x)={\sum }_{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(0)}{n!}{x}^n$

75. Partial Derivative: $\frac{\mathrm{\partial }f}{\mathrm{\partial }x}$

76. Gradient: $\mathrm{\nabla }f=(\frac{\mathrm{\partial }f}{\mathrm{\partial }x},\frac{\mathrm{\partial }f}{\mathrm{\partial }y},\frac{\mathrm{\partial }f}{\mathrm{\partial }z})$

77. Divergence: $\mathrm{\nabla }\cdot \mathbf{F}$

78. Curl: $\mathrm{\nabla }\times \mathbf{F}$

79. Line Integral: ${\int }_C\mathbf{F}\cdot d\mathbf{r}$

80. Surface Integral: ${\iint }_S\mathbf{F}\cdot d\mathbf{S}$

81.