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

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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

  
  
  

  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

  
  
  

  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

  
  
  

  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

  
  
  

  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

  
  
  

  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

  
  
  

  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

  
  
  

  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

  
  
  

  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

  
  
  

  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

  
  
  

  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

  
  
  

  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

  
  
  

  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

  
  
  

  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:

  • Hyperbolic Sine Function
  • 
  • Hyperbolic Cosine Function
  • 
  • Hyperbolic Tangent Function
  • 
  • Hyperbolic Cot Tangent Function
  • 
  • Hyperbolic Secant Function
  • 
  • Hyperbolic Cosech Function cosech(x)
  • 

• 2.10 Differentiation of Hyperbolic Cosine Function cosh(x)

  Equation

  
  
  

  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:

  • Hyperbolic Sine Function
  • 
  • Hyperbolic Cosine Function
  • 
  • Hyperbolic Tangent Function
  • 
  • Hyperbolic Cot Tangent Function
  • 
  • Hyperbolic Secant Function
  • 
  • Hyperbolic Cosech Function cosech(x)
  • 

• 2.11 Differentiation of Hyperbolic Tangent Function tanh(x)

  Equation

  
  
  

  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:

  • Hyperbolic Sine Function
  • 
  • Hyperbolic Cosine Function
  • 
  • Hyperbolic Tangent Function
  • 
  • Hyperbolic Cot Tangent Function
  • 
  • Hyperbolic Secant Function
  • 
  • Hyperbolic Cosech Function cosech(x)
  • 

• 2.12 Differentiation of Hyperbolic Secant Function sech(x)

  Equation

  
  
  

  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:

  • Hyperbolic Sine Function
  • 
  • Hyperbolic Cosine Function
  • 
  • Hyperbolic Tangent Function
  • 
  • Hyperbolic Cot Tangent Function
  • 
  • Hyperbolic Secant Function
  • 
  • Hyperbolic Cosech Function cosech(x)
  • 

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

  Equation

  
  
  

  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:

  • Hyperbolic Sine Function
  • 
  • Hyperbolic Cosine Function
  • 
  • Hyperbolic Tangent Function
  • 
  • Hyperbolic Cot Tangent Function
  • 
  • Hyperbolic Secant Function
  • 
  • Hyperbolic Cosech Function cosech(x)
  • 

• 2.14 Differentiation of Hyperbolic Cosech Function cosech(x)

  Equation

  
  
  

  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:

  • Hyperbolic Sine Function
  • 
  • Hyperbolic Cosine Function
  • 
  • Hyperbolic Tangent Function
  • 
  • Hyperbolic Cot Tangent Function
  • 
  • Hyperbolic Secant Function
  • 
  • Hyperbolic Cosech Function cosech(x)
  •