Subtraction isn’t much harder than addition. Division isn’t much harder than multiplication. Logarithmization isn’t much harder than exponentiation. Then, what fundamentally makes integration so much more difficult than differentiation?
In: Mathematics
Well we know that 3 * 4 = 12. But if I gave you 12 could you tell me which two numbers were multiplied to produce it? That is somewhat analogous to the problem.
Given a function, it is usually relatively straightforward to know what amount the output value is going to change if the input value changes slightly. Taken to the limit, this is what differentiation does.
However given a function and being asked to decompose it into a series of terms that change at the rate at the functions value at all points in that function is not straightforward at all.
Say you had a credit card and your outstanding balance increased by $10 from yesterday to today. Now you ask someone else, “why did that happen?” How would they figure it out? Did you spend $4 on coffee and $6 on a burger? Did you spend $8 on fried chicken and $2 on a soda?
On the other hand if you said “I spent $6 on a hotdog and $4 on a coffee yesterday”, it doesn’t take much to conclude that “you spent $10 in one day”.
Latest Answers