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D

D(f, x)

gives the partial derivative of f with respect to x.

D(f, x, y, ...)

differentiates successively with respect to x, y, etc.

D(f, {x,n})

gives the multiple derivative of order n.

D(f, {{x1, x2, ...}})

gives the vector derivative of f with respect to x1, x2, etc.

Note: the upper case identifier D is different from the lower case identifier d.

See:

Examples

First-order derivative of a polynomial:

>> D(x^3 + x^2, x)
2*x+3*x^2

Second-order derivative:

>> D(x^3 + x^2, {x, 2})
2+6*x

Trigonometric derivatives:

>> D(Sin(Cos(x)), x)
-Cos(Cos(x))*Sin(x)


>> D(Sin(x), {x, 2})
-Sin(x)


>> D(Cos(t), {t, 2})
-Cos(t)

Unknown variables are treated as constant:

>> D(y, x)
0


>> D(x, x)
1


>> D(x + y, x)
1

Derivatives of unknown functions are represented using ‘Derivative’:

>> D(f(x), x)
f'(x)


>> D(f(x, x), x)
Derivative(0,1)[f][x,x]+Derivative(1,0)[f][x,x]

Chain rule:

>> D(f(2*x+1, 2*y, x+y), x)
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]


>> D(f(x^2, x, 2*y), {x,2}, y) // Expand
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(
1,1,1)[f][x^2,x,2*y]+8*x^2*Derivative(2,0,1)[f][x^2,x,2*y]

Compute the gradient vector of a function:

>> D(x ^ 3 * Cos(y), {{x, y}})
{3*x^2*Cos(y),-x^3*Sin(y)}

Hesse matrix:

>> D(Sin(x) * Cos(y), {{x,y}, 2})
{{-Cos(y)*Sin(x),-Cos(x)*Sin(y)},{-Cos(x)*Sin(y),-Cos(y)*Sin(x)}}


>> D(2/3*Cos(x) - 1/3*x*Cos(x)*Sin(x) ^ 2,x)//Expand
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)


>> D(f(#1), {#1,2})
f''(#1)


>> D((#1&)(t),{t,4})
0


>> Attributes(f) = {HoldAll}; Apart(f''(x + x))
f''(2*x)


>> Attributes(f) = {}; Apart(f''(x + x))
f''(2*x)


>> D({#^2}, #)
{2*#1}

Diff, DSolve, Integrate, Limit, ND, NIntegrate

Implementation status

  • ☑ - partially implemented

Github