1. The problem involves understanding and differentiating logarithmic and trigonometric functions, as well as summations and integrals. 2. Key formulas and rules: - Derivative of $\ln x$ is $\frac{1}{x}$. - Derivative of $\tan x$ is $\sec^2 x$. - The sum of squares formula: $\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$. - The product notation $\prod_{k=1}^n k = n!$ (factorial). - The definite integral $\int_a^b f(x)\,dx$ represents the area under $f(x)$ from $a$ to $b$. 3. Differentiation examples: - $\frac{d}{dx} \ln x = \frac{1}{x}$. - $\frac{d}{dx} \tan x = \sec^2 x$. 4. Summation example: - Calculate $\sum_{i=1}^n i^2$ using the formula above. 5. Integral example: - $\int_a^b f(x)\,dx$ depends on the function $f(x)$ and limits $a,b$. 6. The expressions $\Lambda, \Theta, \Gamma, \varphi, \pi, \Omega, \alpha, \beta, \lambda$ are Greek letters often used as variables or constants. 7. Symbols like $\subseteq, \in, \cup, \cap, \supseteq$ denote set relations. 8. The vector notation $\vec{v}$ and unit vector $\hat{x}$ indicate vectors. This collection of symbols and operations is foundational in advanced calculus and algebra. Final answer: Differentiation and summation formulas are applied as above for $\ln x$, $\tan x$, and sums of squares.