====== Differentiate ======
----
====Description====

  * ''Differentiate[var1, var2]'' differentiates expression successively with respect to ''var1'', ''var2'' ...


  * ''Differentiate'' can differentiate tensorial expressions with respect to tensorial variables. ''Differentiate'' takes care about dummies relabelling and symmetries of tensors. The following convention is adopted:\[
 \frac{\partial T_{m_1 \dots m_k}}{\partial T_{n_1 \dots n_k}} = \delta^{n_1}_{m_1} \times \dots \times \delta^{n_k}_{m_k} \,+\, permutations
\]such that r.h.s. has exactly same symmetries as l.h.s.
====Examples====
----
Derivative with respect to ''x'':
<sxh groovy; gutter: false>
println Differentiate['x'] >> 'x**n'.t
</sxh>
<sxh plain; gutter: false>
   > n*x**(n-1)
</sxh>
----
Derivative with respect to ''f_mn'':
\[
\frac{\partial }{\partial f_{mn}} \, \left(\, f_{ab} f_{cd} \, \right) = \delta_a^m \, \delta_b^n \, f_{cd} \,+\, \delta_d^n \, \delta_c^m \, f_{ab}
\]

Redberry code:
<sxh groovy; gutter: false>
println Differentiate['f_mn'] >> 'f_ab*f_cd'.t
</sxh>
<sxh plain; gutter: false>
   > d_a^m*d_b^n * f_cd + d_d^n*d_c^m*f_ab
</sxh>


----
Derivative involving a symbolic function ''f_mn[t_ab]'':
\[
\frac{\partial }{\partial t_{mn}} \, \left(\, t_{ab} f^{ab}(t_{pq}) \, \right) =  f^{mn}(t_{dc}) \,+\, t_{ab} \frac{\partial }{\partial t_{mn}} \left( f^{ab}(t_{dc})\right)
\]

Redberry code:
<sxh groovy; gutter: false>
println Differentiate['t_mn'] >> 't_ab*f^ab[t_pq]'.t
</sxh>
<sxh plain; gutter: false>
   > f^{mn}[t_{dc}] + t_{ab}*f~(1)^{abmn}[t_{dc}]
</sxh>

----
Derivative with respect to ''x_m'' and ''y_m'':
\[
\frac{\partial^4 }{\partial x_{m} \partial x^m \partial y_n \partial y^n} \, \left(\, (x_a y^a)^5 \, \right) =  240\,  (y^b x_b)^3+40\, (y^b x_b)^3 \delta^a_a + 120\, (y_m y^m) (y^b x_b) (x_a x^a)
\]

Redberry code:
<sxh groovy; gutter: false>
def diff = Differentiate['x_m', 'x^m', 'y_a', 'y^a'] & CollectScalars
println diff >> '(x_a*y^a)**5'.t
</sxh>
<sxh plain; gutter: false>
   > 240*(y^b*x_b)**3+40*(y^b*x_b)**3*d^a_a+120*y_m*y^m*y^b*x_b*x_a*x^a
</sxh>

----
Derivative with respect to antisymmetric tensor:
<sxh groovy; gutter: false>
setAntiSymmetric 'R_pq'
println Differentiate['R_mn'] >> 'R_ab'.t
</sxh>
<sxh plain; gutter: false>
   > d_{a}^{m}*d_{b}^{n} - d_{a}^{n}*d_{b}^{m}
</sxh>
----

====Options====
''Differentiate[var1, var2, ..., transformations]'' allows to pass additional ''transformations'' which will be applied after each step of differentiation (for performance reasons):
<sxh groovy; gutter: true>
//setting up symmetries of Riemann tensor
addSymmetry 'R_abcd', -[1, 0, 2, 3].p
addSymmetry 'R_abcd', [2, 3, 0, 1].p

def tensor = 'R^acbd*Sin[R_abcd*R^abcd]'.t
def var1 = 'R^ma_m^b'.t,
    var2 = 'R^mc_m^d'.t
    
def diff1, diff2    
timing { 
  //take second derivative and then simplify
  diff1 = (Differentiate[var2, var1] & ExpandAndEliminate) >> tensor 
}
</sxh>
<sxh plain; gutter: false>
   > Time: 1338 ms.
</sxh>
<sxh groovy; gutter: true; first-line: 14>
timing { 
  //take second derivative and simplify permanently
  diff2 = Differentiate[var2, var1, ExpandAndEliminate] >> tensor
}
</sxh>
<sxh plain; gutter: false>
   > Time: 14 ms.
</sxh>
<sxh groovy; gutter: true; first-line: 18>
assert diff1 == diff2
</sxh>

====See also====
  * Related guides: [[documentation:guide:applying_and_manipulating_transformations]], [[documentation:guide:representation_of_derivatives]], [[documentation:guide:list_of_transformations]]
  * JavaDocs: [[http://api.redberry.cc/redberry/1.1.9/java-api/cc/redberry/core/transformations/DifferentiateTransformation.html| DifferentiateTransformation]]
  * Source code: [[https://bitbucket.org/redberry/redberry/src/tip/core/src/main/java/cc/redberry/core/transformations/DifferentiateTransformation.java|DifferentiateTransformation.java]]