Derivatives and Counting question asked in an interview

Question

The below is a question that had been asked to one of my friends in an interview for admission into a premier management institute of India, the Indian Institute of Management.

Consider x^2. Now consider a series addition of x, x times as in x+x+x+x+....... x times.

Now we can see both x^2 and x+x+x+x+....... x times are same.

Now if we differentiate x^2 with respect to x, we get 2x. And if we differentiate x+x+x+x+....... x times with respect to x, we get 1+1+1+1+..... x times, which equals x.

So why is this difference ?

Also when we look at x+x+x+x+....... x times as x × x, and then if we differentiate this wrt x using the product differentiation rule, we do get 2x as the derivative.

So why this paradox. This was asked in the interview. What could be the answer to this inconsistency?

$(x + x + x \cdots)$ $x$ times is $(x \cdot x)$ and by the product rule $\dfrac{d}{dx}(x \cdot x)$ gives us $x'x + x'x = 2x$ — Skm, Apr 06 '19 at 15:53

score 0 · Answer 1 · answered Apr 06 '19 at 15:33

The problem is $1+1+1...$ ($x$ times) is not the correct derivative of $x+x+x...$ ($x$ times)

This is because the NUMBER OF TIMES x is added to itself must also be differentiated.

In fact, as the slight increase in the x corresponds to a slight increase in the number of times $x$ is multiplied, which does not make sense with this analogy

Derivatives and Counting question asked in an interview

1 Answers1