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

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