Home > Math Shortcuts > Derivative Mathematical Formulas

Derivative Mathematical Formulas


Derivative Mathematical Formulas

 

LIMIT EVALUATION AT +- ∞

  • \lim\limits_{x \to \infty } e^{x} = \infty
  • \lim\limits_{x \to -\infty } e^{x} = 0
  • \lim\limits_{x \to \infty} lnx = \infty
  • \lim\limits_{x \to \infty} \frac{c}{x^{n}} = 0 \left ( n > 0 \right )
  • \lim\limits_{x \to \infty} \frac{x}{\sqrt[x]{x!}} = e
  • \lim\limits_{x \to \infty} \left ( 1 + \frac{k}{x} \right )^{x} = e^{k}, e = 2.71
  • \lim\limits_{x \to \infty} \left ( 1 - \frac{1}{x} \right )^{x} = \frac{1}{e}
  • \lim\limits_{x \to \infty} x \left ( \frac{\sqrt{2\Pi x}}{x!} \right )^{\frac{1}{x}} = e
  • \lim\limits_{x \to \infty} \frac{x!}{x^{x}e^{-x}\sqrt{x}} = \sqrt{2\Pi }
  • \lim\limits_{x \to \infty} log_{a}\left ( 1 + \frac{1}{x} \right )^{x} = log_{a} e

 

PROPERTIES OF LIMITS

  • \lim\limits_{x \to a}[Cf\left ( x \right )] = C \lim_{x \to a}[f\left ( x \right )]
  • \lim\limits_{x \to a}[f\left ( x \right )]^{n} = [\lim_{x \to a} f\left ( x \right )]^{n}

 

 

LIMIT EVALUATION AT ZERO

  • \lim\limits_{x \to 0} a^{x} = 1
  • \lim\limits_{x \to 0} \frac{log_{e}\left ( 1+x \right )}{x} = 1
  • \lim\limits_{x \to 0} \frac{x}{log_{a}\left ( 1+x \right )} = \frac{1}{log_{a}e}
  • \lim\limits_{x \to 0} \frac{a^{x}-1}{x} = ln a, a > 0
  • \lim\limits_{x \to 0} \frac{sin x}{x} = 1
  • \lim\limits_{x \to 0} \frac{tan x}{x} = 1
  • \lim\limits_{x \to 0} \frac{1 - cos x}{x} = 0
  • \lim\limits_{x \to 0} \frac{1 - cos x}{x^{2}} = \frac{1}{2}
  • \lim\limits_{x \to 0} \frac{arcsin x}{x} = 1
  • \lim\limits_{x \to 0} \frac{arctan x}{x} = 1
  • \lim\limits_{x \to 1} \frac{ \left ( arccos x \right )^{2}}{1-x} = 2

 

 

Derivative Definition

  • \frac{d}{dx}\left ( f\left ( x \right ) \right ) = f'\left ( x \right ) = \lim_{h \to 0}\frac{f\left ( x+h \right )-f\left ( x \right )}{h}

 

 

Basic Properties

  • \left ( cf\left ( x \right ) \right )' = c\left ( f'\left ( x \right ) \right )
  • \left ( f\left ( x \right )\pm g\left ( x \right ) \right )' = f'\left ( x \right )\pm g'\left ( x \right )
  • \frac{d}{dx}\left ( c \right )=0

 

 

Mean Value Theorem

  • f'\left ( c \right ) = \frac{f\left ( b \right )-f\left ( a \right )}{b-a}

 

 

Quotient Rule

  • \frac{d}{dx}\left ( \frac{f\left ( x \right )}{g\left ( x \right )} \right ) = \frac{f'\left ( x \right )g\left ( x \right )-f\left ( x \right )g'\left ( x \right )}{\left [ g\left ( x \right ) \right ]^{2}}

 

 

Power Rule

  • \frac{d}{dx}\left ( x^{n} \right ) = nx^{n-1}

 

 

Chain Rule

  • \frac{d}{dx}\left ( f\left ( g\left ( x \right ) \right ) \right ) = f'\left ( g\left ( x \right ) \right )g'\left ( x \right )

 

 

Limit Evaluation Method

  • \lim\limits_{x \to -3}\frac{x^{2}-x-12}{x^{2}+3x} = \lim\limits_{x \to -3}\frac{\left ( x+3 \right )\left ( x-4 \right )}{x\left ( x+3 \right )} = \lim\limits_{x \to -3}\frac{\left ( x-4 \right )}{x} = \frac{7}{3}

 

 

L’Hopital’s Rule

  • \lim\limits_{x \to a}\frac{f\left ( x \right )}{g\left ( x \right )} = \frac{0}{0} or \frac{\pm \infty }{\pm \infty } then \lim\limits_{x \to a}\frac{f\left ( x \right )}{g\left ( x \right )} = \lim\limits_{x \to a}\frac{f'\left ( x \right )}{g'\left ( x \right )}

 

 

Common Derivatives

  • \frac{d}{dx}\left ( x \right ) = 1
  • \frac{d}{dx}\left ( sin x \right ) = cos x
  • \frac{d}{dx}\left ( cos x \right ) = -sin x
  • \frac{d}{dx}\left ( tan x \right ) = sec^{2} x
  • \frac{d}{dx}\left ( sec x \right ) = sec x tan x
  • \frac{d}{dx}\left ( csc x \right ) = -csc x cot x
  • \frac{d}{dx}\left ( cot x \right ) = -csc^{2} x
  • \frac{d}{dx}\left ( sin^{-1} x \right ) = \frac{1}{\sqrt{1-x^{2}}}
  • \frac{d}{dx}\left ( cos^{-1} x \right ) = -\frac{1}{\sqrt{1-x^{2}}}
  • \frac{d}{dx}\left ( tan^{-1} x \right ) = -\frac{1}{1+x^{2}}
  • \frac{d}{dx}\left ( a^{x} \right ) = a^{x} ln\left ( a \right )
  • \frac{d}{dx}\left ( e^{x} \right ) = e^{x}
  • \frac{d}{dx}\left ( ln\left ( x \right ) \right ) = \frac{1}{x}, x>0
  • \frac{d}{dx}\left ( ln\left | x \right | \right ) = \frac{1}{x}
  • \frac{d}{dx}\left ( log_{a}\left ( x \right ) \right ) = \frac{1}{x ln\left ( a \right )}

 

 

Chain Rule and Other Examples

  • \frac{d}{dx}\left ( \left [ f\left ( x \right ) \right ]^{n} \right ) = n\left [ f\left ( x \right ) \right ]^{n-1}f'\left ( x \right )
  • \frac{d}{dx}\left ( e^{f\left ( x \right )} \right ) = f'\left ( x \right )e^{f\left ( x \right )}
    \frac{d}{dx}\left ( ln\left [ f\left ( x \right ) \right ] \right ) = \frac{f'\left ( x \right )}{f\left ( x \right )}
  • \frac{d}{dx}\left ( sin\left [ f\left ( x \right ) \right ] \right ) = f'\left ( x \right )cos\left [ f\left ( x \right ) \right ]
  • \frac{d}{dx}\left ( cos\left [ f\left ( x \right ) \right ] \right ) = -f'\left ( x \right )sin\left [ f\left ( x \right ) \right ]
  • \frac{d}{dx}\left ( tan\left [ f\left ( x \right ) \right ] \right ) = f'\left ( x \right )sec^{2}\left [ f\left ( x \right ) \right ]
  • \frac{d}{dx}\left ( sec\left [ f\left ( x \right ) \right ] \right ) = f'\left ( x \right )sec\left [ f\left ( x \right ) \right ]tan\left [ f\left ( x \right ) \right ]
  • \frac{d}{dx}\left ( tan^{-1}\left [ f\left ( x \right ) \right ] \right ) = \frac{f'\left ( x \right )}{1+\left [ f\left ( x \right ) \right ]^{2}}
  • \frac{d}{dx}\left ( f\left ( x \right )^{g\left ( x \right )} \right ) = f\left ( x \right )^{g\left ( x \right )}\left ( \frac{g\left ( x \right )f'\left ( x \right )}{f\left ( x \right )}+ln \left ( f\left ( x \right ) \right )g'\left ( x \right ) \right )

 

 

Properties of Limit

  • \lim\limits_{x \to a}\left [ f\left ( x \right )\pm g\left ( x \right ) \right ] = \lim\limits_{x \to a} f\left ( x \right )\pm \lim\limits_{x \to a} g\left ( x \right )
  • \lim\limits_{x \to a}\left [ f\left ( x \right ) g\left ( x \right ) \right ] = \lim\limits_{x \to a} f\left ( x \right ) \lim\limits_{x \to a} g\left ( x \right )
  • \lim\limits_{x \to a}\left [ \frac{f\left ( x \right )}{g\left ( x \right )} \right ] = \frac{\lim\limits_{x \to a}f\left ( x \right )}{\lim\limits_{x \to a}g\left ( x \right )} if \lim_{x \to a}g\left ( x \right )\neq 0

 

 

 

We provide few shortcut tricks on this topic. Please visit this page to get updates on more Math Shortcut Tricks. You can also like our facebook page to get updates.

If You Have any question regarding this topic then please do comment on below section. You can also send us message on facebook.


Leave a Reply