Difference between revisions of "Derivative/Formulas"
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== List of formulas == | == List of formulas == | ||
− | {| style="margin: 1em auto 1em auto" | + | {| style="margin: 1em auto 1em auto; height:1000px" |
− | | | + | | <math>\frac d{dx}(cf(x)) = c\left(\frac d{dx} f(x)\right)</math> where <math>c</math> is a constant |
|- | |- | ||
− | | | + | | <math>(f(x) + g(x))' = f'(x) + g'(x)</math> |
|- | |- | ||
− | | | + | | <math>(f(x)-g(x))'=f'(x)-g'(x)</math> |
|- | |- | ||
− | | | + | | <math>\left(u(x)\times v(x)\right)'=u(x)v'(x)+u'(x)v(x)</math> |
|- | |- | ||
− | | | + | | <math>\left(\frac{u(x)}{v(x)}\right)' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}</math> |
|- | |- | ||
− | | | + | | <math>(f(g(x)))' = f'(g(x))g'(x)</math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} x^n = n x^{n-1}</math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} (f(x))^n =n f(x)^{n-1} f'(x)</math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} \sin x = \cos x</math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} \cos x = -\sin x</math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} \tan x = \sec^2 x</math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} \sec x = \sec x \tan x</math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} \csc x = -\csc x\cot x</math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} \cot x = -\csc^2 x</math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} e^x = e^x</math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} a^x = (\ln a) a^x</math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} \ln x = \frac 1x</math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} \log_b x =\frac{\log_b e}{x} </math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} \arcsin x = \frac 1{\sqrt{1-x^2}}</math> |
|- | |- | ||
− | | | + | | <math>\frac d{dx} \arccos x = -\frac 1{\sqrt{1-x^2}}</math> |
+ | |- | ||
+ | | <math>\frac d{dx} \arctan x = \frac 1{1+x^2}</math> | ||
+ | |- | ||
+ | | <math>\frac d{dx} \mathrm{arcsec \ } x = \frac 1{\lvert x \rvert \sqrt{x^2-1}}</math> | ||
+ | |- | ||
+ | | <math>\frac d{dx} \mathrm{arccsc \ } x = - \frac 1{x\sqrt{x^2 - 1}}</math> | ||
+ | |- | ||
+ | | <math>\frac d{dx} \mathrm{arccot \ } x = - \frac 1{1+x^2}</math> | ||
+ | |} | ||
+ | |||
+ | == Notation == | ||
+ | The following are commonly recognized notations for expressing the [[derivative]] of a [[function]]. | ||
+ | |||
+ | {| class="wikitable" style="text-align:center; margin: 1em auto 1em auto; height:600px; width:300px" | ||
+ | | colspan="2" | '''Euler's notation''' | ||
+ | |- | ||
+ | | First derivative || <math>D_xf(x)</math> or <math>Du</math> | ||
+ | |- | ||
+ | | Second derivative || <math>D_x^2f(x)</math> or <math>D^2u</math> | ||
+ | |- | ||
+ | | Third derivative || <math>D_x^3f(x)</math> or <math>D^3u</math> | ||
+ | |- | ||
+ | | <math>n</math>th derivative || <math>D_x^nf(x)</math> or <math>D^nu</math> | ||
+ | |- | ||
+ | | colspan="2" | '''Lagrange's notation''' | ||
+ | |- | ||
+ | | First derivative || <math>f'(x)</math> | ||
+ | |- | ||
+ | | Second derivative || <math>f''(x)</math> | ||
+ | |- | ||
+ | | Third derivative || <math>f'''(x)</math> | ||
+ | |- | ||
+ | | <math>n</math>th derivative || <math>f^{(n)}(x)</math> | ||
+ | |- | ||
+ | | colspan="2" | '''Leibniz's notation''' | ||
+ | |- | ||
+ | | First derivative || <math>\frac{dy}{dx}</math> | ||
+ | |- | ||
+ | | Second derivative || <math>\frac{d^2y}{dx^2}</math> | ||
+ | |- | ||
+ | | <math>n</math>th derivative || <math>\frac{d^ny}{dx^n}</math> | ||
+ | |- | ||
+ | | colspan="2" | '''Newton's notation''' | ||
+ | |- | ||
+ | | First derivative || <math>\dot{x}</math> | ||
+ | |- | ||
+ | | Second derivative || <math>\ddot{x}</math> | ||
|} | |} | ||
Latest revision as of 16:23, 11 March 2022
List of formulas
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Notation
The following are commonly recognized notations for expressing the derivative of a function.
Euler's notation | |
First derivative | ![]() ![]() |
Second derivative | ![]() ![]() |
Third derivative | ![]() ![]() |
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Lagrange's notation | |
First derivative | ![]() |
Second derivative | ![]() |
Third derivative | ![]() |
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Leibniz's notation | |
First derivative | ![]() |
Second derivative | ![]() |
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Newton's notation | |
First derivative | ![]() |
Second derivative | ![]() |