29 Calculus
29.1 Fundamental limits
| \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) | \(\lim_{x \to 0} \frac{\ln (1 + x) }{x} = 1\) | \(\lim_{x \to 0} \frac{e^x -1 }{x} = 1\) |
| \(\lim_{x \to 0} \left(1 + \frac{1}{x}\right)^{x} = e\) | \(\lim_{x \to 0} \frac{\log_{a} (1 + x) }{x} = \log_{a}e\) | \(\lim_{x \to 0} \frac{a^x -1 }{x} = \ln a\) |
29.2 Derivatives
| Function \(f(x)\) | Derivative \(f^{\prime}(x)\) |
|---|---|
| \(y = a, \; a \in \mathbb{R}\) | \(y^{\prime} = 0\) |
| \(y = x^n, \; n \in \mathbb{N}\) | \(y^{\prime} = nx^{n-1}\) |
| \(y = x^\alpha, \; \alpha \in \mathbb{R}\) | \(y^{\prime} = \alpha x^{\alpha-1}\) |
| \(y = x^{\frac{1}{n}}, \; n > 0\) | \(y^{\prime} = \frac{1}{n} x^{\frac{1}{n} - 1}\) |
| \(y = \sin x\) | \(y^{\prime} = \cos x\) |
| \(y = \cos x\) | \(y^{\prime} = - \sin x\) |
| \(y = \tan x\) | \(y^{\prime} = \frac{1}{\cos^2 x} = 1 + \tan^2 x\) |
| \(y = \cot x\) | \(y^{\prime} = - \frac{1}{\sin^2 x} = - (1 + \cot^2 x)\) |
| \(y = \arcsin x\) | \(y^{\prime} = \frac{1}{\sqrt{1 - x^2}}\) |
| \(y = \arccos x\) | \(y^{\prime} = - \frac{1}{\sqrt{1 - x^2}}\) |
| \(y = \arctan x\) | \(y^{\prime} = \frac{1}{1 + x^2}\) |
| \(y = \text{arccot}\ x\) | \(y^{\prime} = - \frac{1}{1 + x^2}\) |
| \(y = a^x\) | \(y^{\prime} = a^x \ln a\) |
| \(y = e^x\) | \(y^{\prime} = e^x\) |
| \(y = \log_a x\) | \(y^{\prime} = \frac{1}{x \ln a}\) |
| \(y = \ln x\) | \(y^{\prime} = \frac{1}{x}\) |
| \(f(x) = c \cdot g(x)\) | \(f^{\prime}(x) = c \cdot g^{\prime}(x)\) |
| \(f(x) = g(x) + s(x)\) | \(f^{\prime}(x) = g^{\prime}(x) + s^{\prime}(x)\) |
| \(f(x) = \frac{1}{g(x)}\) | \(f^{\prime}(x) = -\frac{g^{\prime}(x)}{g(x)^2}\) |
Derivative of the product: \[ [f(x) \cdot g(x)]^{\prime} (x) = f^{\prime}(x) g(x) + f(x) g^{\prime}(x) \text{.} \]
Derivative of the ratio: \[ \left[\frac{f(x)}{g(x)}\right]^{\prime}(x) = \frac{f^{\prime}(x) g(x) - f(x) g^{\prime}(x)}{[g(x)]^2} \text{.} \]
Derivative of the composition: \[ [f(g(x))]^{\prime}(x) = g^{\prime}(x) \cdot f^{\prime}(g(x)) \text{.} \]
For example \(f(g(x)) = \ln(1 + 2x)\), then \(f(x) = \ln(x)\) and \(g(x) = 1 + 2x\), hence \[ [\ln(1 + 2x)]^{\prime}(x) = [1 + 2x]^{\prime}(x) \cdot [\ln(x)]^{\prime}(1 + x) \text{.} \]
29.2.1 Taylor series
\[ f(x) = \sum_{n = 1}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n = f(a) + f^{\prime}(a)(x - a) + \frac{f^{\prime\prime}(a)}{2!}(x-a)^2 +\dots \tag{29.1}\]
29.3 Integrals
| Immediate | General |
|---|---|
| \(\int x^{n} dx = \frac{x^{n+1}}{n+1} + c\) | \(\int f(x)^{n} dx = \frac{f(x)^{n+1}}{n+1} + c\) |
| \(\int \frac{1}{x} dx = \log(x) + c\) | \(\int \frac{f^{\prime}(x)}{f(x)} dx = \log(f(x)) + c\) |
| \(\int a^{x} dx = \log_{a}(e) + c\) | \(\int a^{f(x)} f^{\prime}(x) dx = a^{f(x)} \log_{a}(e) + c\) |
| \(\int e^{x} dx = e^{x} + c\) | \(\int e^{x} f^{\prime}(x) dx = e^{f(x)} + c\) |
| \(\int \sin(x) dx = -\cos(x) + c\) | \(\int \sin(f(x)) f^{\prime}(x) dx = -\cos(f(x)) + c\) |
| \(\int \cos(x) dx = \sin(x) + c\) | \(\int \cos(f(x))f^{\prime}(x) dx = \sin(f(x)) + c\) |
| \(\int \frac{1}{\cos(x)^2} dx = \tan(x) + c\) | \(\int \frac{f^{\prime}(x)}{\cos(f(x))^2} dx = \tan(f(x)) + c\) |
| \(\int \frac{1}{\sin(x)^2} dx = \cot(x) + c\) | \(\int \frac{f^{\prime}(x)}{\sin(f(x))^2} dx = \cot(f(x)) + c\) |
| \(\int \frac{1}{\sqrt{1-x^2}} dx = \arcsin(x) + c\) | \(\int \frac{f^{\prime}(x)}{\sqrt{1-f(x)^2}} dx = \arcsin(f(x)) + c\) |
| \(\int \frac{1}{\sqrt{a^2-x^2}} dx = \arcsin(\frac{x}{a}) + c\) | \(\int \frac{f^{\prime}(x)}{\sqrt{a^2-f(x)^2}} dx = \arcsin(\frac{f(x)}{a}) + c\) |
| \(\int \frac{1}{\sqrt{1+x^2}} dx = \arctan(x) + c\) | \(\int \frac{f^{\prime}(x)}{\sqrt{1+f(x)^2}} dx = \arctan(f(x)) + c\) |
29.3.1 Fundamental theorem
\[ f(b) - f(a) = \int_{a}^{b} f^{\prime}(x)dx \iff \int f^{\prime}(x)dx = f(x) + C \text{.} \]
29.3.2 Integration by parts
\[ \int_{a}^{b} {\color{red}{f({\color{black}{x}})}} {\color{blue}{g^{\prime}({\color{black}{x}})}} dx = \left[{\color{red}{f({\color{black}{x}})}} {\color{blue}{g({\color{black}{x}})}} \right]_{x = a}^{x = b} - \int_{a}^{b} {\color{red}{f^{\prime}({\color{black}{x}})}} {\color{blue}{g({\color{black}{x}})}} dx \text{,} \] or in compact form: \[ \int_{a}^{b} {\color{red}{f({\color{black}{x}})}}d{\color{blue}{g({\color{black}{x}})}} = \left[{\color{red}{f({\color{black}{x}})}} {\color{blue}{g({\color{black}{x}})}} \right]_{x = a}^{x = b} - \int_{a}^{b} {\color{blue}{g({\color{black}{x}})}} d{\color{red}{f({\color{black}{x}})}} dx \text{.} \]
29.4 Useful relations
- Newton binomial
\[ (a + b)^n = \sum_{k = 0}^{\infty} \binom{n}{k} a^{n-k} b^{k} \text{.} \]