Difference between revisions of "Derivative/Formulas"

Latest revision as of 15:23, 11 March 2022

List of formulas

$\frac d{dx}(cf(x)) = c\left(\frac d{dx} f(x)\right)$ where $c$ is a constant

$(f(x) + g(x))' = f'(x) + g'(x)$

$(f(x)-g(x))'=f'(x)-g'(x)$

$\left(u(x)\times v(x)\right)'=u(x)v'(x)+u'(x)v(x)$

$\left(\frac{u(x)}{v(x)}\right)' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$

$(f(g(x)))' = f'(g(x))g'(x)$

$\frac d{dx} x^n = n x^{n-1}$

$\frac d{dx} (f(x))^n =n f(x)^{n-1} f'(x)$

$\frac d{dx} \sin x = \cos x$

$\frac d{dx} \cos x = -\sin x$

$\frac d{dx} \tan x = \sec^2 x$

$\frac d{dx} \sec x = \sec x \tan x$

$\frac d{dx} \csc x = -\csc x\cot x$

$\frac d{dx} \cot x = -\csc^2 x$

$\frac d{dx} e^x = e^x$

$\frac d{dx} a^x = (\ln a) a^x$

$\frac d{dx} \ln x = \frac 1x$

$\frac d{dx} \log_b x =\frac{\log_b e}{x}$

$\frac d{dx} \arcsin x = \frac 1{\sqrt{1-x^2}}$

$\frac d{dx} \arccos x = -\frac 1{\sqrt{1-x^2}}$

$\frac d{dx} \arctan x = \frac 1{1+x^2}$

$\frac d{dx} \mathrm{arcsec \ } x = \frac 1{\lvert x \rvert \sqrt{x^2-1}}$

$\frac d{dx} \mathrm{arccsc \ } x = - \frac 1{x\sqrt{x^2 - 1}}$

$\frac d{dx} \mathrm{arccot \ } x = - \frac 1{1+x^2}$

Notation

The following are commonly recognized notations for expressing the derivative of a function.

Euler's notation
First derivative	$D_xf(x)$ or $Du$
Second derivative	$D_x^2f(x)$ or $D^2u$
Third derivative	$D_x^3f(x)$ or $D^3u$
$n$ th derivative	$D_x^nf(x)$ or $D^nu$
Lagrange's notation
First derivative	$f'(x)$
Second derivative	$f''(x)$
Third derivative	$f'''(x)$
$n$ th derivative	$f^{(n)}(x)$
Leibniz's notation
First derivative	$\frac{dy}{dx}$
Second derivative	$\frac{d^2y}{dx^2}$
$n$ th derivative	$\frac{d^ny}{dx^n}$
Newton's notation
First derivative	$\dot{x}$
Second derivative	$\ddot{x}$

@@ Line 2: / Line 2: @@
 {| style="margin: 1em auto 1em auto; height:1000px"
-| <math>\frac d{dx}(cf(x)) = c\left(\frac d{dx} f(x)\right)</math>
+| <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>
@@ Line 44: / Line 44: @@
 | <math>\frac d{dx} \arctan x = \frac 1{1+x^2}</math>
 |-
-| <math>\frac d{dx} \mathrm{arcsec \ } x = \frac 1{\mid x \mid\sqrt{x^2-1}}</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} \arccot x = - \frac 1{1+x^2}</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>
 |}

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Difference between revisions of "Derivative/Formulas"

Latest revision as of 15:23, 11 March 2022

List of formulas

Notation

See also