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There are options in FullSimplify like TransformationFunctions or ComplexityFunction which might help in various cases. In our case a simple and direct approach would be defining a list of rules. Here is an example:

![enter image description here][1]

rules =
  { c_ Sum[n a[n] c_^(n-1), {n, 0, Infinity}] :> Sum[n c^n a[n], {n, 0, Infinity}],
    α_ Sum[a[n] c_^n, {n, 0, Infinity}] + Sum[n a[n] c_^n, {n, 0, Infinity}] :>
    Sum[(α + n) a[n] c^n, {n, 0, Infinity}]};

Let's define an appropriate function for TransformationFunctions applying rules to an expression:

 tf[expr_] := expr /. rules  

and now FullSimplify with tf does the expected transformation:

FullSimplify[ D[y, x], TransformationFunctions -> {Automatic, tf}]//TraditionalForm

![enter image description here][2]

alternatively one can do this:

FullSimplify[ D[y, x]] //. rules

Note that here we had to use ReplaceRepeated (//.) while applying rules in TransformationFunctions it was quite sufficient to play with ReplaceAll (/.). [1]: https://i.sstatic.net/3LZMI.gif [2]: https://i.sstatic.net/OjChv.gif [3]: https://i.sstatic.net/piBiw.gif [4]: https://i.sstatic.net/aehdX.gif

In our case a simple and direct approach would be defining a list of rules. Here is an example:

![enter image description here][1]

rules =
  { c_ Sum[n a[n] c_^(n-1), {n, 0, Infinity}] :> Sum[n c^n a[n], {n, 0, Infinity}],
    α_ Sum[a[n] c_^n, {n, 0, Infinity}] + Sum[n a[n] c_^n, {n, 0, Infinity}] :>
    Sum[(α + n) a[n] c^n, {n, 0, Infinity}]};

Let's define an appropriate function for TransformationFunctions applying rules to an expression:

 tf[expr_] := expr /. rules  

and now FullSimplify with tf does the expected transformation:

FullSimplify[ D[y, x], TransformationFunctions -> {Automatic, tf}]//TraditionalForm

![enter image description here][2]

alternatively one can do this:

FullSimplify[ D[y, x]] //. rules

Note: Applying rules in TransformationFunctions it was quite sufficient to play with ReplaceAll (/.) while in the latter case we had to use rules repeatedly i.e. with ReplaceRepeated (//.) . [1]: https://i.sstatic.net/3LZMI.gif [2]: https://i.sstatic.net/OjChv.gif [3]: https://i.sstatic.net/piBiw.gif [4]: https://i.sstatic.net/aehdX.gif

There are options in FullSimplify like TransformationFunctions or ComplexityFunction which might help in various cases. In our case a simple and direct approach would be defining a list of rules. Here is an example:

rules =
  { c_ Sum[n a[n] c_^(n-1), {n, 0, Infinity}] :> Sum[n c^n a[n], {n, 0, Infinity}],
    α_ Sum[a[n] c_^n, {n, 0, Infinity}] + Sum[n a[n] c_^n, {n, 0, Infinity}] :>
    Sum[(α + n) a[n] c^n, {n, 0, Infinity}]};

Let's define an appropriate function for TransformationFunctions:

 tf[expr_] := expr //. rules  

and now FullSimplify with tf does the expected transformation:

FullSimplify[ D[y, x], TransformationFunctions -> {Automatic, tf}]//TraditionalForm

alternatively one can do this:

FullSimplify[ D[y, x]] //. rules

There are options in FullSimplify like TransformationFunctions or ComplexityFunction which might help in various cases. In our case a simple and direct approach would be defining a list of rules. Here is an example:

![enter image description here][1]

rules =
  { c_ Sum[n a[n] c_^(n-1), {n, 0, Infinity}] :> Sum[n c^n a[n], {n, 0, Infinity}],
    α_ Sum[a[n] c_^n, {n, 0, Infinity}] + Sum[n a[n] c_^n, {n, 0, Infinity}] :>
    Sum[(α + n) a[n] c^n, {n, 0, Infinity}]};

Let's define an appropriate function for TransformationFunctions applying rules to an expression:

 tf[expr_] := expr /. rules  

and now FullSimplify with tf does the expected transformation:

FullSimplify[ D[y, x], TransformationFunctions -> {Automatic, tf}]//TraditionalForm

![enter image description here][2]

alternatively one can do this:

FullSimplify[ D[y, x]] //. rules

Note that here we had to use ReplaceRepeated (//.) while applying rules in TransformationFunctions it was quite sufficient to play with ReplaceAll (/.). [1]: https://i.sstatic.net/3LZMI.gif [2]: https://i.sstatic.net/OjChv.gif [3]: https://i.sstatic.net/piBiw.gif [4]: https://i.sstatic.net/aehdX.gif

There are options in FullSimplify like TransformationFunctions or ComplexityFunction and they might help in various cases. Here is an example of possible rules:

rules = 
  List[
       RuleDelayed[
         Times[ Pattern[c, Blank[]], 
                Sum[ Times[n, a[n], Power[Pattern[c, Blank[]], Plus[-1, n]]], 
                     List[n, 0, DirectedInfinity[1]]]], 
         Sum[ Times[n, Power[c, n], a[n]], List[n, 0, Infinity]]
                 ],                 
       RuleDelayed[
         Plus[ Times[ Pattern[\[Alpha], Blank[]], 
                      Sum[ Times[a[n], Power[Pattern[c, Blank[]], n]], 
                           List[n, 0, DirectedInfinity[1]]]], 
               Sum[ Times[n, a[n], Power[Pattern[c, Blank[]], n]], 
                    List[n, 0, DirectedInfinity[1]]]], 
               Sum[Times[Plus[\[Alpha], n], Power[c, n], a[n]], List[n, 0, Infinity]]]      
      ];

Let's define appropriate TransformationFunctions

alternatively one can do this:

 FullSimplify[ D[y, x]] //. rules

There are options in FullSimplify like TransformationFunctions or ComplexityFunction which might help in various cases. In our case a simple and direct approach would be defining a list of rules. Here is an example:

rules =
  { c_ Sum[n a[n] c_^(n-1), {n, 0, Infinity}] :> Sum[n c^n a[n], {n, 0, Infinity}],
    α_ Sum[a[n] c_^n, {n, 0, Infinity}] + Sum[n a[n] c_^n, {n, 0, Infinity}] :>
    Sum[(α + n) a[n] c^n, {n, 0, Infinity}]};

Let's define an appropriate function for TransformationFunctions:

 tf[expr_] := expr //. rules  

and now FullSimplify with tf does the expected transformation:

FullSimplify[ D[y, x], TransformationFunctions -> {Automatic, tf}]//TraditionalForm

alternatively one can do this:

FullSimplify[ D[y, x]] //. rules