For this question I understand the first line of the solution which they have obtained from the definition but how have they simplified each term to get to the second line from the first line? The step to go from the second to third line I understand as they have grouped terms 1 and 3 and 2 and 4 to get the final line. But what do they mean by repeat this time on the second entry? Is the first entry consisting of the terms fg and so the second entry consists of the terms hk? Also why is there a need to do this on seperate entries, why cant we do this in one go from the definition? Also in the final step on the first entry they haven't shown LHS equals RHS so how do we get to the full answer.
Any help would be much appreciated.
Going from the first line to the second, they've just used the product rule on the first factor of each term.
When they say "repeat this for the second entry", this is what they mean. The computation they showed gave an equation where, on the RHS, they "pulled out" a function from the first slot of the bracket. They want you to do an analogous thing to "pull out" a factor from the second slot (this will mean doing the product rule on the second factors in line 1 and grouping terms).
As you say, we could do all of this in one fell swoop rather than slot by slot.
Can you take it from here?