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‎symja_android_library/doc/functions/D.md

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```
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> gives the partial derivative of `f` with respect to `x`.
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```
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D(f, x, y, ...)
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```
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> differentiates successively with respect to `x`, `y`, etc.
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```
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D(f, {x,n})
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```
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> gives the `n`th derivative of `f` with respect to `x`.
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> gives the multiple derivative of order `n`.
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```
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D(f, {{x1, x2, ...}})
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```
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> gives the vector derivative of `f` with respect to `x1`, `x2`, etc.
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**Note**: the upper case identifier `D` is different from the lower case identifier `d`.
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### Examples
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```
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>> D(Sin(x),x)
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Cos(x)
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### Examples
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First-order derivative of a polynomial:
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```
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>> D(x^3 + x^2, x)
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2*x+3*x^2
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```
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Second-order derivative:
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```
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>> D(x^3 + x^2, {x, 2})
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2+6*x
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```
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Trigonometric derivatives:
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```
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>> D(Sin(Cos(x)), x)
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-Cos(Cos(x))*Sin(x)
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>> D(Sin(x), {x, 2})
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-Sin(x)
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>> D(Cos(t), {t, 2})
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-Cos(t)
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```
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Unknown variables are treated as constant:
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```
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>> D(y, x)
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0
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>> D(x, x)
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1
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>> D(x + y, x)
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1
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```
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Derivatives of unknown functions are represented using 'Derivative':
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```
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>> D(f(x), x)
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f'(x)
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>> D(f(x, x), x)
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Derivative(0,1)[f][x,x]+Derivative(1,0)[f][x,x]
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```
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Chain rule:
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```
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>> D(f(2*x+1, 2*y, x+y)
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2*Derivative(1,0,0)[f][1+2*x,2*y,x+y]+Derivative(0,0,1)[f][1+2*x,2*y,x+y]
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>> D(f(x^2, x, 2*y), {x,2}, y) // Expand
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2*Derivative(0,2,1)[f][x^2,x,2*y]+4*Derivative(1,0,1)[f][x^2,x,2*y]+8*x*Derivative(
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1,1,1)[f][x^2,x,2*y]+8*x^2*Derivative(2,0,1)[f][x^2,x,2*y]
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```
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Compute the gradient vector of a function:
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```
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>> D(x^5,{x,2})
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20*x^3
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```
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>> D(x ^ 3 * Cos(y), {{x, y}})
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{3*x^2*Cos(y),-x^3*Sin(y)}
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```
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Hesse matrix:
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```
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>> D(Sin(x) * Cos(y), {{x,y}, 2})
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{{-Cos(y)*Sin(x),-Cos(x)*Sin(y)},{-Cos(x)*Sin(y),-Cos(y)*Sin(x)}}
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>> D(2/3*Cos(x) - 1/3*x*Cos(x)*Sin(x) ^ 2,x)//Expand
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1/3*x*Sin(x)^3-1/3*Sin(x)^2*Cos(x)-2/3*Sin(x)-2/3*x*Cos(x)^2*Sin(x)
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>> D(f(#1), {#1,2})
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f''(#1)
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>> D((#1&)(t),{t,4})
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0
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>> Attributes(f) = {HoldAll}; Apart(f''(x + x))
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f''(2*x)
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>> Attributes(f) = {}; Apart(f''(x + x))
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f''(2*x)
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>> D({#^2}, #)
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{2*#1}
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```

‎symja_android_library/doc/functions/Derivative.md

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## Derivative
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```
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Derivative(n)[f]
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```
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> represents the `n`-th derivative of the function `f`.
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```
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Derivative(n1, n2, n3,...)[f]
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```
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> represents a multivariate derivative.
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### Examples
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```
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>> Derivative(1)[Sin]
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Cos(#1)&
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>> Derivative(3)[Sin]
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-Cos(#1)&
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>> Derivative(2)[# ^ 3&]
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6*(#1&)
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```
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`Derivative` can be entered using `'`:
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```
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>> Sin'(x)
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Cos(x)
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>> (# ^ 4&)''
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12*(#1^2&)
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>> f'(x) // FullForm
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"Derivative(1)[f][x]"
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```
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The `0`th derivative of any expression is the expression itself:
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```
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>> Derivative(0,0,0)[a+b+c]
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a+b+c
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```
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Unknown derivatives:
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```
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>> Derivative(2, 1)[h]
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Derivative(2,1)[h]
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>> Derivative(2, 0, 1, 0)[h(g)]
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Derivative(2,0,1,0)[h(g)]
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```

‎symja_android_library/doc/index.md

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* [DeleteDuplicates](functions/DeleteDuplicates.md)
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* [Denominator](functions/Denominator.md)
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* [Depth](functions/Depth.md)
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* [Derivative](functions/Derivative.md)
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* [DesignMatrix](functions/DesignMatrix.md)
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* [Det](functions/Det.md)
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* [DiagonalMatrix](functions/DiagonalMatrix.md)

‎symja_android_library/matheclipse-core/src/main/resources/D.md

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```
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> gives the partial derivative of `f` with respect to `x`.
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```
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D(f, x, y, ...)
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```
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> differentiates successively with respect to `x`, `y`, etc.
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```
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D(f, {x,n})
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```
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> gives the `n`th derivative of `f` with respect to `x`.
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> gives the multiple derivative of order `n`.
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```
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D(f, {{x1, x2, ...}})
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```
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> gives the vector derivative of `f` with respect to `x1`, `x2`, etc.
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**Note**: the upper case identifier `D` is different from the lower case identifier `d`.
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### Examples
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```
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>> D(Sin(x),x)
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Cos(x)
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### Examples
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First-order derivative of a polynomial:
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```
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>> D(x^3 + x^2, x)
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2*x+3*x^2
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```
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Second-order derivative:
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```
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>> D(x^3 + x^2, {x, 2})
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2+6*x
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```
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Trigonometric derivatives:
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```
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>> D(Sin(Cos(x)), x)
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-Cos(Cos(x))*Sin(x)
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>> D(Sin(x), {x, 2})
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-Sin(x)
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>> D(Cos(t), {t, 2})
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-Cos(t)
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```
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Unknown variables are treated as constant:
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```
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>> D(y, x)
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0
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>> D(x, x)
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1
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>> D(x + y, x)
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1
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```
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Derivatives of unknown functions are represented using 'Derivative':
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```
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>> D(f(x), x)
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f'(x)
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>> D(f(x, x), x)
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Derivative(0,1)[f][x,x]+Derivative(1,0)[f][x,x]
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```
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Chain rule:
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```
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>> D(f(2*x+1, 2*y, x+y)
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2*Derivative(1,0,0)[f][1+2*x,2*y,x+y]+Derivative(0,0,1)[f][1+2*x,2*y,x+y]
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>> D(f(x^2, x, 2*y), {x,2}, y) // Expand
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2*Derivative(0,2,1)[f][x^2,x,2*y]+4*Derivative(1,0,1)[f][x^2,x,2*y]+8*x*Derivative(
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1,1,1)[f][x^2,x,2*y]+8*x^2*Derivative(2,0,1)[f][x^2,x,2*y]
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```
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Compute the gradient vector of a function:
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```
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>> D(x^5,{x,2})
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20*x^3
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```
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>> D(x ^ 3 * Cos(y), {{x, y}})
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{3*x^2*Cos(y),-x^3*Sin(y)}
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```
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Hesse matrix:
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```
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>> D(Sin(x) * Cos(y), {{x,y}, 2})
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{{-Cos(y)*Sin(x),-Cos(x)*Sin(y)},{-Cos(x)*Sin(y),-Cos(y)*Sin(x)}}
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>> D(2/3*Cos(x) - 1/3*x*Cos(x)*Sin(x) ^ 2,x)//Expand
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1/3*x*Sin(x)^3-1/3*Sin(x)^2*Cos(x)-2/3*Sin(x)-2/3*x*Cos(x)^2*Sin(x)
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>> D(f(#1), {#1,2})
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f''(#1)
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>> D((#1&)(t),{t,4})
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0
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>> Attributes(f) = {HoldAll}; Apart(f''(x + x))
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f''(2*x)
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>> Attributes(f) = {}; Apart(f''(x + x))
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f''(2*x)
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>> D({#^2}, #)
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{2*#1}
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```

‎symja_android_library/matheclipse-core/src/main/resources/Derivative.md

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## Derivative
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```
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Derivative(n)[f]
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```
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> represents the `n`-th derivative of the function `f`.
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```
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Derivative(n1, n2, n3,...)[f]
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```
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> represents a multivariate derivative.
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### Examples
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```
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>> Derivative(1)[Sin]
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Cos(#1)&
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>> Derivative(3)[Sin]
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-Cos(#1)&
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>> Derivative(2)[# ^ 3&]
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6*(#1&)
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```
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`Derivative` can be entered using `'`:
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```
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>> Sin'(x)
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Cos(x)
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>> (# ^ 4&)''
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12*(#1^2&)
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>> f'(x) // FullForm
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"Derivative(1)[f][x]"
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```
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The `0`th derivative of any expression is the expression itself:
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```
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>> Derivative(0,0,0)[a+b+c]
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a+b+c
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```
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Unknown derivatives:
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```
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>> Derivative(2, 1)[h]
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Derivative(2,1)[h]
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>> Derivative(2, 0, 1, 0)[h(g)]
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Derivative(2,0,1,0)[h(g)]
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```

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