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 usefull 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 :
  $${\frac{{\partial^2 (x.y^2.z^3+x.y.z)}}{{\partial x\partial z}}}$$
  Input :
  diff(x*y ^2*z^3+x*y*z,x,z)
  Output :
  y^2*3*z^2+y
• Compute :
  $${\frac{{\partial^3 (x.y^2.z^3+x.y.z)}}{{\partial x\partial^2 z}}}$$
  Input :
  diff(x*y ^2*z^3+x*y*z,x,z,z)
  Or input :
  diff(x*y ^2*z^3+x*y*z,x,z$2)
  Output :
  y^2*3*2*z
• Compute the third derivative of :
  $${\frac{{1}}{{x^2+2}}}$$
  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 $${\frac{{\partial^2(f)}}{{\partial x\partial y}}}$$ and
  diff(f,[x, y]) returns [$${\frac{{\partial(f)}}{{\partial x}}}$$,$${\frac{{\partial (f)}}{{\partial y}}}$$]
• 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)