code: def test_eval_partial_derivative_expr1():
      tau, alpha = symbols("tau alpha")
  
      # this is only some special expression
      # tested: vector derivative
      # tested: scalar derivative
      # tested: tensor derivative
      base_expr1 = (
          A(i) * H(-i, j) + A(i) * A(-i) * A(j) + tau**alpha * A(j)
      )
  
      tensor_derivative = PartialDerivative(
          base_expr1, H(k, m)
      )._perform_derivative()
      vector_derivative = PartialDerivative(
          base_expr1, A(k)
      )._perform_derivative()
      scalar_derivative = PartialDerivative(
          base_expr1, tau
      )._perform_derivative()
  
      assert (
          tensor_derivative
          - A(L_0)
          * L.metric(-L_0, -L_1)
          * L.delta(L_1, -k)
          * L.delta(j, -m)
      ) == 0
  
      assert (
          vector_derivative
          - (
              tau**alpha * L.delta(j, -k)
              + L.delta(L_0, -k) * A(-L_0) * A(j)
              + A(L_0) * L.metric(-L_0, -L_1) * L.delta(L_1, -k) * A(j)
              + A(L_0) * A(-L_0) * L.delta(j, -k)
              + L.delta(L_0, -k) * H(-L_0, j)
          )
      ).expand() == 0
  
      assert (
          vector_derivative.contract_metric(L.metric).contract_delta(
              L.delta
          )
          - (
              tau**alpha * L.delta(j, -k)
              + A(L_0) * A(-L_0) * L.delta(j, -k)
              + H(-k, j)
              + 2 * A(j) * A(-k)
          )
      ).expand() == 0
  
      assert (
          scalar_derivative - alpha * 1 / tau * tau**alpha * A(j) == 0
      )
  
imports: from sympy import sin, cos
  from sympy.testing.pytest import raises
  from sympy.tensor.toperators import PartialDerivative
  from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead, tensor_heads
  from sympy.core.numbers import Rational
  from sympy.core.symbol import symbols
  from sympy.matrices.dense import diag
  from sympy.tensor.array import Array
  from sympy.core.random import randint
label_desc: Use image node_id 13 for calling a global function with example usage: test_eval_partial_derivative_expr1() without return types