keywords:quotient rule, chain rule, Leibniz rule
Derivatives
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| General Rules | |
|---|---|
| Derivative of a constant | $ \frac{d}{dx}\left( c \right) = 0, \ \text{ for any constant }c $ |
| $ \frac{d}{dx}\left( c x \right) = c, \ \text{ for any constant }c $ | |
| Linearity | $ \frac{d}{dx}\left( c_1 u_1+c_2 u_2 \right) = c_1 \frac{d}{dx}\left( u_1 \right)+c_2 \frac{d}{dx}\left( u_2 \right), \ \text{ for any constants }c_1, c_2 $ |
| Quotient rule | $ \frac{d}{dx} ( \frac{u}{v} ) = \frac{v ( \frac{du}{dx} ) - u ( \frac{dv}{dx} )}{v^2} $ |
| Exponent rule | $ \frac{d}{dx} ( u^n ) = n u^{n-1} \frac{du}{dx} $ |
| Chain rule | $ \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} $ |
| $ \frac{du}{dx} = \frac{1}{\frac{dx}{du}} $ | |
| $ \frac{dy}{dx} = \frac{dy}{du}/\frac{dx}{du} $ | |
| Leibnitz Rule for Successive Derivatives of a Product | |
| first order | $ \frac{d}{dx}\left( u v \right)= u \frac{dv }{dx} + v \frac{du }{dx} $ |
| second order | $ \frac{d^2}{dx^2}\left( u v \right)= u \frac{d^2v }{dx^2} + 2\frac{du }{dx}\frac{dv }{dx}+ v \frac{d^2u }{dx^2} $ |
| third order | $ \frac{d^3}{dx^3}\left( u v \right)= u \frac{d^3v }{dx^3} + 3 \frac{du }{dx}\frac{d^2v }{dx^2}+ 3 \frac{du^2 }{dx^2}\frac{d v }{dx}+ v \frac{d^3u }{dx^3} $ |
| n-th order | $ \frac{d^n}{dx^n}\left( u v \right)= u \frac{d^n v }{dx^n} + \left( \begin{array}{cc}n \\ 1 \end{array}\right) \frac{du }{dx}\frac{d^{n-1}v }{dx^{n-1}} + \left( \begin{array}{cc}n \\ 2 \end{array}\right) \frac{d^2u}{dx^2}\frac{d^{n-2}v }{dx^{n-2}}+ \ldots + v \frac{d^n u }{dx^n} $ |
| Derivatives of trigonometric functions | ||
|---|---|---|
| $ \frac {d}{dx} \sin u = \cos u \frac{du}{dx} $ | ||
| $ \frac {d}{dx} \cos u = - \sin u \frac{du}{dx} $ | ||
| $ \frac {d}{dx} \tan u = \frac{1}{\cos^2 u} \frac{du}{dx} $ | ||
| $ \frac {d}{dx} \cot u = - \frac{1}{\sin^2 u} \frac{du}{dx} $ | ||
| $ \frac {d}{dx} \frac{1}{\cos u} = \frac{\tan u}{\cos u} \frac{du}{dx} $ | ||
| $ \frac {d}{dx} \frac{1}{\sin u} = - \frac{\cot u}{\sin u} \frac{du}{dx} $ | ||
| $ \frac {d}{dx} \arcsin u = \frac{1}{\sqrt{1-u^2}} \frac{du}{dx} \qquad ( - \frac{\pi}{2} < \arcsin u < \frac{\pi}{2} ) $ | ||
| $ \frac {d}{dx} \arccos u = - \frac{1}{\sqrt{1-u^2}} \frac{du}{dx} \qquad ( 0 < \arccos u < \pi ) $ | ||
| $ \frac {d}{dx} \arctan u = \frac{1}{1+u^2} \frac{du}{dx} \qquad ( - \frac{\pi}{2} < \arctan u < \frac{\pi}{2} ) $ | ||
| $ \frac {d}{dx} \arccot u = - \frac{1}{1+u^2} \frac{du}{dx} \qquad ( 0 < \arccot u < \pi ) $ | ||
| Derivatives of exponential and logarithm functions | ||
| $ \frac{d}{dx} \log_a u = \frac{log_a e}{u} \frac{du}{dx} \qquad a \neq 0,1 $ | ||
| $ \frac{d}{dx} \ln u = \frac{d}{dx} log_e u = \frac{1}{u} \frac{du}{dx} $ | ||
| $ \frac{d}{dx} a^u = a^u \ln a \frac{du}{dx} $ | ||
| $ \frac{d}{dx} e^u = e^u \frac{du}{dx} $ | ||
| $ \frac{d}{dx} u^v = \frac{d}{dx} e^{v ln u} = e^{v ln u} \frac {d}{dx} [ v ln u ] = v u^{v-1} \frac{du}{dx} + u^v ln u \frac{dv}{dx} $ | ||
| Derivatives of hyperbolic functions | ||
| $ \frac{d}{dx} \sinh u = \cosh u \frac{du}{dx} $ | ||
| $ \frac{d}{dx} \cosh u = \sinh u \frac{du}{dx} $ | ||
| $ \frac{d}{dx} \tanh u = \frac{1}{\cosh^2 u} \frac{du}{dx} $ | ||
| $ \frac{d}{dx} \coth u = - \frac{1}{\sinh^2 u} \frac{du}{dx} $ | ||
| $ \frac{d}{dx} \frac{1}{\cosh u} = - \frac{\tanh u}{\cosh u} \frac{du}{dx} $ | ||
| $ \frac{d}{dx} \frac{1}{\sinh u} = - \frac{\coth u}{\sinh u} \frac{du}{dx} $ | ||
| $ \frac{d}{dx}\ \operatorname{arsinh}\ u = \frac{1}{\sqrt{u^2+1}} \frac{du}{dx} $ | ||
| $ \frac{d}{dx}\ \operatorname{arcosh}\ u = \frac{1}{\sqrt{u^2-1}} \frac{du}{dx} $ | ||
| $ \frac{d}{dx}\ \operatorname{artanh}\ u = \frac{1}{1-u^2} \frac{du}{dx} \qquad ( \ -1 < u < 1 \ ) $ | ||
| $ \frac{d}{dx}\ \operatorname{arcoth}\ u = \frac{1}{1-u^2} \frac{du}{dx} \qquad ( \ u > 1 \ or \ u < -1 \ ) $ | ||
