Derivative and partial derivative

### 5.19.5 Derivative and partial derivative

diff or derive may have one or two arguments to compute a first order derivative (or first order partial derivative) of an expression or of a list of expressions, or several arguments to compute the n-th partial derivative of an expression or list of expressions.

#### Derivative and first order partial derivative : diff derive deriver

diff (or derive) takes two arguments : an expression and a variable (resp. a vector of variable names) (see several variable functions in 5.59). If only one argument is provided, the derivative is taken with respect to x
diff (or derive) returns the derivative (resp. a vector of derivatives) of the expression with respect to the variable (resp. with respect to each variable) given as second argument.
Examples :

- Compute :

  ∂ (x y2 z3+x y z)
  ∂ z

  Input :
  diff(x*y`^`2*z`^`3+x*y*z,z)
  Output :
  x*y`^`2*3*z`^`2+x*y
- Compute the 3 first order partial derivatives of x*y2*z3+x*y*z.
  Input :
  diff(x*y`^`2*z`^`3+x*y,[x,y,z])
  Output :
  [y`^`2*z`^`3+y*z, x*2*y*z`^`3+x*z, x*y`^`2*3*z`^`2+x*y]

#### Derivative and n-th order partial derivative : diff derive deriver

derive (or diff) may take more than two arguments : an expression and the names of the derivation variables (each variable may be followed by $n to indicate the number n of derivations).
diff returns the partial derivative of the expression with respect to the variables given after the first argument.

The notation $ is useful if you want to derive k times with respect to the same variable, instead of entering k times the same variable name, one enters the variable name followed by $k, for example x$3 instead of (x,x,x). Each variable may be followed by a $, for example diff(exp(x*y),x$3,y$2,z) is the same as diff(exp(x*y),x,x,x,y,y,z)
Examples

- Compute :

  ∂2 (x y2 z3+x y z)
  ∂ x∂ z

  Input :
  diff(x*y`^`2*z`^`3+x*y*z,x,z)
  Output :
  y`^`2*3*z`^`2+y
- Compute :

  ∂3 (x y2 z3+x y z)
  ∂ x∂2 z

  Input :
  diff(x*y`^`2*z`^`3+x*y*z,x,z,z)
  or :
  diff(x*y`^`2*z`^`3+x*y*z,x,z$2)
  Output :
  y`^`2*3*2*z
- Compute the third derivative of :

  1
  x2+2

  Input :
  normal(diff((1)/(x`^`2+2),x,x,x))
  or :
  normal(diff((1)/(x`^`2+2),x$3))
  Output :
  (-24*x`^`3+48*x)/(x`^`8+8*x`^`6+24*x`^`4+32*x`^`2+16)

Remark

- Note the difference between diff(f,x,y) and diff(f,[x,y]) :
  diff(f,x,y) returns ∂2(f)/∂ x∂ y and
  diff(f,[x,y]) returns [∂(f)/∂ x,∂ (f)/∂ y]
- Never define a derivative function with f1(x):=diff(f(x),x). Indeed, x would mean two different things Xcas is unable to deal with: the variable name to define the f1 function and the differentiation variable. The right way to define a derivative is either with function_diff or:
  f1:=unapply(diff(f(x),x),x)