List of Calculus Formulas Latex Code (First Part: Limits, Differentiation)

rockingdingo 2022-12-04 #calculus #differentiation #Math

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In this blog, we will summarize the latex code for basic calculus formulas, including Limits, Differentiation and Integration. For limits formulas, we will cover sections including: L'Hospital Rule, Limits of Power, Limits of xln(x), Limits of x^{n}/n!. For differentiation formulas, we will cover Differentiation of Polynomial Function, Chain Rule for Differentiation, Differentiation of Multiplication, Differentiation of Division, Differentiation of Trigonometric formulas (sin, cos, tan, sec), and Differentiation of Hyperbolic formulas (sinh, cosh, tanh, sech, coth, cosech).

1. Limits

1.1 L'Hospital Rule

1.2 Limits of Power

1.3 Limits of xln(x)

1.4 Limits of x^{n}/n!

2. Differentiation

2.1 Differentiation of Polynomial Function

2.2 Chain Rule for Differentiation

2.3 Differentiation of Multiplication

2.4 Differentiation of Division

2.5 Differentiation of Sine Function sin(x)

2.6 Differentiation of Cosine Function cos(x)

2.7 Differentiation of Tangent Function tan(x)

2.8 Differentiation of Secant Function sec(x)

2.9 Differentiation of Hyperbolic Sine Function sinh(x)

2.10 Differentiation of Hyperbolic Cosine Function cosh(x)

2.11 Differentiation of Hyperbolic Tangent Function tanh(x)

2.12 Differentiation of Hyperbolic Secant Function sech(x)

2.13 Differentiation of Hyperbolic Cot-Tangent Function coth(x)

2.14 Differentiation of Hyperbolic Cosech Function cosech(x)

1. Limits

1.1 L'Hospital Rule

Equation

$\lim_{x\rightarrow c} \frac{f(x)}{g(x)}=\frac{f^{'}(c)}{g^{'}(c)},\text{given} f(c)=g(c)=0$

Latex Code

\lim_{x\rightarrow c} \frac{f(x)}{g(x)}=\frac{f^{'}(c)}{g^{'}(c)},\text{given} f(c)=g(c)=0

Explanation

As x approaches c of "f-ofâ??x over g-ofâ??x", the limit value of \lim_{x\rightarrow c} equals the the limit as x approaches c of "f-dash-ofâ??x over g-dash-ofâ??x".
1.2 Limits of Power

Equation

$\lim_{n \rightarrow \infty}(1+\frac{x}{n})^{n} \rightarrow e^{x}$

Latex Code
```
        	 \lim_{n \rightarrow \infty}(1+\frac{x}{n})^{n} \rightarrow e^{x}
        
```
Explanation

The limit of power n of (1+x/n) equals to e^{x} as n approaches infinity.
1.3 Limits of xln(x)

Equation

$\lim_{n \rightarrow \infty}(1+\frac{x}{n})^{n} \rightarrow e^{x}$

Latex Code
```
        	 \lim_{x \rightarrow 0}x\ln(x)=0
        
```
Explanation

The limit of xln(x) equals to 0 as x approaches 0.
1.4 Limits of x^{n}/n!

Equation

$\lim_{n \rightarrow \infty}\frac{x^{n}}{n!}=0$

Latex Code
```
        	 \lim_{n \rightarrow \infty}\frac{x^{n}}{n!}=0
        
```
Explanation

The limit of x^{n}/n! equals to 0 as n approaches infinity.

2. Differentiation

2.1 Differentiation of Polynomial function

Equation

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{d}}{\mathrm{d} x} f(x)=f^{'}(x) \\\frac{\mathrm{d}}{\mathrm{d} x} (ax^{n})=anx^{n-1} \\\frac{\mathrm{d}}{\mathrm{d} x} (ax)=a \\\frac{\mathrm{d}}{\mathrm{d} x} (a)=0 \\\frac{\mathrm{d}}{\mathrm{d} x} (ax^{m} + bx^{n})=amx^{m-1} + bnx^{n-1}$

Latex Code

        \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{d}}{\mathrm{d} x} f(x)=f^{'}(x) \\\frac{\mathrm{d}}{\mathrm{d} x} (ax^{n})=anx^{n-1} \\\frac{\mathrm{d}}{\mathrm{d} x} (ax)=a \\\frac{\mathrm{d}}{\mathrm{d} x} (a)=0 \\\frac{\mathrm{d}}{\mathrm{d} x} (ax^{m} + bx^{n})=amx^{m-1} + bnx^{n-1}

Explanation

The differentiation of polynomial function d(ax^{n})/dx equals to anx^{n-1}.

2.2 Chain Rule for Differentiation

Equation

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{d} y}{\mathrm{d} z} \times \frac{\mathrm{d} z}{\mathrm{d} x}$

Latex Code

            \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{d} y}{\mathrm{d} z} \times \frac{\mathrm{d} z}{\mathrm{d} x}

Explanation

Chain rule for Differentiation.

2.3 Differentiation of Multiplication

Equation

$\frac{\mathrm{d}}{\mathrm{d} x}(uv)=u \frac{\mathrm{d} v}{\mathrm{d} x} + v \frac{\mathrm{d} u}{\mathrm{d} x}$

Latex Code

            \frac{\mathrm{d}}{\mathrm{d} x}(uv)=u \frac{\mathrm{d} v}{\mathrm{d} x} + v \frac{\mathrm{d} u}{\mathrm{d} x}

Explanation

Differentiation of Multiplication.

2.4 Differentiation of Division

Equation

$\frac{\mathrm{d}}{\mathrm{d} x}(\frac{u}{v})=\frac{v \frac{\mathrm{d} u}{\mathrm{d} x} - u \frac{\mathrm{d} v}{\mathrm{d} x}}{v^{2}}$

Latex Code

            \frac{\mathrm{d}}{\mathrm{d} x}(\frac{u}{v})=\frac{v \frac{\mathrm{d} u}{\mathrm{d} x} - u \frac{\mathrm{d} v}{\mathrm{d} x}}{v^{2}}

Explanation

Differentiation of Division.

2.5 Differentiation of Sine Function sin(x)

Equation

$\frac{\mathrm{d}}{\mathrm{d} x}(\sin(x))=\cos(x)$

Latex Code
```
            \frac{\mathrm{d}}{\mathrm{d} x}(\sin(x))=\cos(x)
        
```
Explanation

Differentiation of Sine Function sin(x)
2.6 Differentiation of Cosine Function cos(x)

Equation

$\frac{\mathrm{d}}{\mathrm{d} x}(\cos(x))=-\sin(x)$

Latex Code
```
            \frac{\mathrm{d}}{\mathrm{d} x}(\cos(x))=-\sin(x)
        
```
Explanation

Differentiation of Cosine Function cos(x)
2.7 Differentiation of Tangent Function tan(x)

Equation

$\frac{\mathrm{d}}{\mathrm{d} x}(\tan(x))=\sec^{2}(x)$

Latex Code
```
            \frac{\mathrm{d}}{\mathrm{d} x}(\tan(x))=\sec^{2}(x)
        
```
Explanation

Differentiation of Tangent Function tan(x)
2.8 Differentiation of Secant Function sec(x)

Equation

$\frac{\mathrm{d}}{\mathrm{d} x}(\sec(x))=\sec(x)\tan(x)$

Latex Code
```
            \frac{\mathrm{d}}{\mathrm{d} x}(\sec(x))=\sec(x)\tan(x)
        
```
Explanation

Differentiation of Sec Function sec(x)
2.9 Differentiation of Hyperbolic Sine Function sinh(x)

Equation

$\frac{\mathrm{d}}{\mathrm{d} x}(\sinh(x))=\cosh(x)$

Latex Code
Differentiation of Hyperbolic Function
```
            \frac{\mathrm{d}}{\mathrm{d} x}(\sinh(x))=\cosh(x)
        
```
Hyperbolic function
```
            \sinh(x)=\frac{e^{x}-e^{-x}}{2} \\ 
            \cosh(x)=\frac{e^{x}+e^{-x}}{2} \\
            \tanh(x)= \frac{\sinh(x)}{\cosh(x)} \\
            \text{sech}(x)= \frac{1}{\cosh(x)} \\
            \text{cosech}(x)= \frac{1}{\sinh(x)}
        
```
Explanation

Differentiation of Hyperbolic Function sinh(x). The calculation of hyperbolic is defined as blow:
2.10 Differentiation of Hyperbolic Cosine Function cosh(x)

Equation

$\frac{\mathrm{d}}{\mathrm{d} x}(\cosh(x))=\sinh(x)$

Latex Code
Differentiation of Hyperbolic Function
```
            \frac{\mathrm{d}}{\mathrm{d} x}(\cosh(x))=\sinh(x)
        
```
Hyperbolic function
```
            \sinh(x)=\frac{e^{x}-e^{-x}}{2} \\ 
            \cosh(x)=\frac{e^{x}+e^{-x}}{2} \\
            \tanh(x)= \frac{\sinh(x)}{\cosh(x)} \\
            \text{sech}(x)= \frac{1}{\cosh(x)} \\
            \text{cosech}(x)= \frac{1}{\sinh(x)}
        
```
Explanation

Differentiation of Hyperbolic Function. The calculation of hyperbolic is defined as blow:
2.11 Differentiation of Hyperbolic Tangent Function tanh(x)

Equation

$\frac{\mathrm{d}}{\mathrm{d} x}(\tanh(x))=\text{sech}^{2} (x)$

Latex Code
Differentiation of Hyperbolic Function
```
            \frac{\mathrm{d}}{\mathrm{d} x}(\tanh(x))=\text{sech}^{2} (x)
        
```
Hyperbolic function
```
            \sinh(x)=\frac{e^{x}-e^{-x}}{2} \\ 
            \cosh(x)=\frac{e^{x}+e^{-x}}{2} \\
            \tanh(x)= \frac{\sinh(x)}{\cosh(x)} \\
            \text{sech}(x)= \frac{1}{\cosh(x)} \\
            \text{cosech}(x)= \frac{1}{\sinh(x)}
        
```
Explanation

Differentiation of Hyperbolic Function. The calculation of hyperbolic is defined as blow:
2.12 Differentiation of Hyperbolic Secant Function sech(x)

Equation

$\frac{\mathrm{d}}{\mathrm{d} x}(\text{sech}(x))=-\text{sech}(x) \tanh(x)$

Latex Code
Differentiation of Hyperbolic Function
```
            \frac{\mathrm{d}}{\mathrm{d} x}(\text{sech}(x))=-\text{sech}(x) \tanh(x)
        
```
Hyperbolic function
```
            \sinh(x)=\frac{e^{x}-e^{-x}}{2} \\ 
            \cosh(x)=\frac{e^{x}+e^{-x}}{2} \\
            \tanh(x)= \frac{\sinh(x)}{\cosh(x)} \\
            \text{sech}(x)= \frac{1}{\cosh(x)} \\
            \text{cosech}(x)= \frac{1}{\sinh(x)}
        
```
Explanation

Differentiation of Hyperbolic Function. The calculation of hyperbolic is defined as blow:
2.13 Differentiation of Hyperbolic Cot-Tangent Function coth(x)

Equation

$\frac{\mathrm{d}}{\mathrm{d} x}(\text{coth}(x))=-\text{cosech}^{2} (x)$

Latex Code
Differentiation of Hyperbolic Function
```
            \frac{\mathrm{d}}{\mathrm{d} x}(\text{coth}(x))=-\text{cosech}^{2} (x)
        
```
Hyperbolic function
```
            \sinh(x)=\frac{e^{x}-e^{-x}}{2} \\ 
            \cosh(x)=\frac{e^{x}+e^{-x}}{2} \\
            \tanh(x)= \frac{\sinh(x)}{\cosh(x)} \\
            \text{sech}(x)= \frac{1}{\cosh(x)} \\
            \text{cosech}(x)= \frac{1}{\sinh(x)}
        
```
Explanation

Differentiation of Hyperbolic Function. The calculation of hyperbolic is defined as blow:
2.14 Differentiation of Hyperbolic Cosech Function cosech(x)

Equation

$\frac{\mathrm{d}}{\mathrm{d} x}(\text{cosech}(x))=-\text{cosech}(x)\text{coth}(x)$

Latex Code
Differentiation of Hyperbolic Function
```
            \frac{\mathrm{d}}{\mathrm{d} x}(\text{cosech}(x))=-\text{cosech}(x)\text{coth}(x)
        
```
Hyperbolic function
```
            \sinh(x)=\frac{e^{x}-e^{-x}}{2} \\ 
            \cosh(x)=\frac{e^{x}+e^{-x}}{2} \\
            \tanh(x)= \frac{\sinh(x)}{\cosh(x)} \\
            \text{sech}(x)= \frac{1}{\cosh(x)} \\
            \text{cosech}(x)= \frac{1}{\sinh(x)}
        
```
Explanation

Differentiation of Hyperbolic Function. The calculation of hyperbolic is defined as blow:

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Navigation

1. Limits

1.1 L'Hospital Rule

1.2 Limits of Power

1.3 Limits of xln(x)

1.4 Limits of x^{n}/n!

2. Differentiation

2.1 Differentiation of Polynomial function

2.2 Chain Rule for Differentiation

2.3 Differentiation of Multiplication

2.4 Differentiation of Division

2.5 Differentiation of Sine Function sin(x)

2.6 Differentiation of Cosine Function cos(x)

2.7 Differentiation of Tangent Function tan(x)

2.8 Differentiation of Secant Function sec(x)

2.9 Differentiation of Hyperbolic Sine Function sinh(x)

2.10 Differentiation of Hyperbolic Cosine Function cosh(x)

2.11 Differentiation of Hyperbolic Tangent Function tanh(x)

2.12 Differentiation of Hyperbolic Secant Function sech(x)

2.13 Differentiation of Hyperbolic Cot-Tangent Function coth(x)

2.14 Differentiation of Hyperbolic Cosech Function cosech(x)