Differentiation

differentiation Leibniz

$\frac{ dy }{ dx }$

\frac{ dy }{ dx }

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This represents the derivative of y with respect to x expressed in fractional form.

differentiation Leibniz 2

$\frac{ \mathrm{ d } y }{ \mathrm{ d } x }$

\frac{ \mathrm{ d } y }{ \mathrm{ d } x }

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This is the Roman font representation of "d" used in the previous example.

differentiation Leibniz 3

$\require{physics} \dv{y}{x}$

\require{physics} \dv{y}{x}

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By using the "physics" extension, we can simplify the notation of "d" in Roman font. The "dv" is derived from "derivative." When we write it with a single curly bracket, only the denominator part remains.

nth differentiation Leibniz

$\frac{ d^n y }{ dx^n }$

\frac{ d^n y }{ dx^n }

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This represents the expression of the nth derivative of y with respect to x in fractional form.

nth differentiation Leibniz 2

$\require{physics} \dv[n]{f}{x}$

\require{physics} \dv[n]{f}{x}

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By using the "physics" extension, we can simplify the notation for the nth derivative.

differentiation Leibniz at a point

$\left. \frac{dy}{dx} \right|_{x=a}$

\left. \frac{dy}{dx} \right|_{x=a}

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A long vertical line drawn on the right side represents the evaluation of the expression at x=a. It is indicated using a subscript notation.

differentiation Leibniz at a point 2

$\require{physics} \eval{\dv{y}{x}}_{x=a}$

\require{physics} \eval{\dv{y}{x}}_{x=a}

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By using the "physics" extension, we can simplify the expression for the value of the derivative at x=a.

differentiation Lagrange

$f'$

f'

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With the use of the apostrophe symbol (') in notation, we can represent the derivative.

second differentiation Lagrange

$f^{\prime\prime}$

f^{\prime\prime}

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If using two consecutive apostrophe symbols ('') does not display correctly, the notation "\prime" can be used instead to represent the derivative.

nth differentiation Lagrange

$f^{ ( n ) }$

f^{ ( n ) }

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differentiation Euler

$Df$

Df

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differentiation Euler 2

$D_x f$

D_x f

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nth differentiation Euler

$D^n f$

D^n f

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differentiation Newton

$\dot{y} = \frac{dy}{dt}$

\dot{y} = \frac{dy}{dt}

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There is also a method of representing differentiation by placing a dot above a character.

4th differentiation Newton

$\ddddot{ y } = \frac{ d^4 y }{ dt^4 }$

\ddddot{ y } = \frac{ d^4 y }{ dt^4 }

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Increasing "d" increases the number of dots.

differentiation sample

$\begin{eqnarray} f'(x) = \frac{ df }{ dx } = \lim_{ \Delta x \to 0 } \frac{ f(x + \Delta x) - f(x) }{ \Delta x } \end{eqnarray}$

\begin{eqnarray}
f'(x)
 = \frac{ df }{ dx }
 = \lim_{ \Delta x \to 0 } \frac{ f(x + \Delta x) - f(x) }{ \Delta x }
\end{eqnarray}

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partial differentiation

$\frac{ \partial f }{ \partial x }$

\frac{ \partial f }{ \partial x }

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It is derived from the partial derivative.

2nd partial differentiation

$\frac{ \partial }{ \partial y } \frac{ \partial }{ \partial x } z$

\frac{ \partial }{ \partial y } \frac{ \partial }{ \partial x } z

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nth partial differentiation

$\frac{ \partial^n f}{ \partial x^n }$

\frac{ \partial^n f}{ \partial x^n }

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partial differentiation 2

$\require{physics} \pdv{f}{x}$

\require{physics} \pdv{f}{x}

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By using the "physics" extension, we can simplify the notation for expressing partial derivatives.

2nd partial differentiation 2

$\require{physics} \pdv{f}{x}{y}$

\require{physics} \pdv{f}{x}{y}

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By utilizing the "physics" extension, we can simplify the notation for representing second-order partial derivatives.

nth partial differentiation 2

$\require{physics} \pdv[n]{f}{x}$

\require{physics} \pdv[n]{f}{x}

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By using the "physics" extension, we can simplify the notation for representing nth-order partial derivatives. Simply write "n" inside brackets to denote the order of the partial derivative.

partial differentiation simple

$f_x$

f_x

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2nd partial differentiation simple

$f_{ xy }$

f_{ xy }

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del

$\nabla f$

\nabla f

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lapracian

$\Delta f$

\Delta f

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lapracian sample

$\begin{eqnarray} \Delta \varphi = \nabla^2 \varphi = \frac{ \partial^2 \varphi }{ \partial x^2 } + \frac{ \partial^2 \varphi }{ \partial y^2 } + \frac{ \partial^2 \varphi }{ \partial z^2 } \end{eqnarray}$

\begin{eqnarray}
\Delta \varphi
 = \nabla^2 \varphi
 = \frac{ \partial^2 \varphi }{ \partial x^2 }
   + \frac{ \partial^2 \varphi }{ \partial y^2 }
   + \frac{ \partial^2 \varphi }{ \partial z^2 }
\end{eqnarray}

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first derivative test table

$\begin{array}{c|ccccc} x & \cdots & -1 & \cdots & 1 & \cdots \\ \hline f’(x) & + & 0 & – & 0 & + \\ \hline f(x) & \nearrow & e & \searrow & -e & \nearrow \end{array}$

\begin{array}{c|ccccc}
  x     & \cdots & -1 & \cdots & 1 & \cdots \\ 
  \hline
  f’(x) & + & 0 & – & 0 & + \\ 
  \hline
  f(x)  & \nearrow & e & \searrow & -e & \nearrow
\end{array}

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