## 3.5.1 Differentiation

 D[f, x] partial derivative D[f, x, y, ... ] multiple derivative D[f, {x, n}] n derivative D[f, x, NonConstants -> {, , ... }] with the taken to depend on

Partial differentiation operations.
This gives .
 In[1]:=  D[x^n, x]
 Out[1]=
This gives the third derivative.
 In[2]:=  D[x^n, {x, 3}]
 Out[2]=
You can differentiate with respect to any expression that does not involve explicit mathematical operations.
 In[3]:=  D[ x[1]^2 + x[2]^2, x[1] ]
 Out[3]=
D does partial differentiation. It assumes here that y is independent of x.
 In[4]:=  D[x^2 + y^2, x]
 Out[4]=
If y does in fact depend on x, you can use the explicit functional form y[x]. Section 3.5.4 describes how objects like y'[x] work.
 In[5]:=  D[x^2 + y[x]^2, x]
 Out[5]=
Instead of giving an explicit function y[x], you can tell D that y implicitly depends on x. D[y, x, NonConstants->{y}] then represents , with y implicitly depending on x.
 In[6]:=  D[x^2 + y^2, x, NonConstants -> {y}]
 Out[6]=

 D[f, {{, , ... }}] the gradient of a scalar function f D[f, {{, , ... }, 2}] the Hessian matrix for f D[f, {{, , ... }, n}] the order Taylor series coefficient D[{, , ... }, {{, , ... }}] the Jacobian for a vector function f

Vector derivatives.
This gives the gradient of the function .
 In[7]:=  D[x^2 + y^2, {{x, y}}]
 Out[7]=
This gives the Hessian.
 In[8]:=  D[x^2 + y^2, {{x, y}, 2}]
 Out[8]=
This gives the Jacobian for a vector function.
 In[9]:=  D[{x^2 + y^2, x y}, {{x, y}}]
 Out[9]=

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