Problem: Find the derivative of

Solution: The function is given by a formula, so we use rules to differentiate. The last operation performed is the arcsine, so we face a composition with arcsine as the outer function that has to be differentiated first (using the chain rule).

Now the last operation is division, so we apply the quotient rule.

Now we use linearity and elementary derivatives to conclude the calculation.

What is the domain of this derivative? We start with the domain of the given function. Since x² + 1 > 0, we need not worry about the fraction. Thus the only restriction comes from the arcsine, which requires that the fraction inside is between −1 and 1 included. We try to solve both equations.

We see that both inequalities are true for all real numbers x, so

D( f ) = ℝ.

Is anything different with the derivative? There is a condition in the first fraction that rules out the case x² = 1, so the formula does not give derivative for x equal to −1 and 1. However, this does not automatically mean that there is no derivative there, it just says that the approach through rules does not help. Thus we explore derivative at these two points as a separate problem, we try the approach through one-sided derivatives. Since the given function f is continuous everywhere, we can find the one-sided derivatives using a limit (see the appropriate theorem in Theory - MVT - Derivative and limit).

However, before we start calculating, we will try to simplify the formula we just obtained. We know that the root of something squared is not the something, but absolute value of the something. We use it to rewrite the derivative and then try to get rid of absolute values.

Now we are ready to calculate the one-sided derivatives. We start with x = −1:

Since the one-sided derivatives do not agree, there is no derivative at x = −1. The same conclusion applies to x = 1, since there we have

Thus we have an interesting situation: The given function is defined on the whole real line, it is given by a seemingly innocent expression that does not feature absolute values, and yet the derivative has a smaller domain:

D( f ′) = (−∞,−1) ∪ (−1,1) ∪ (1,∞).

Things like this occasionally happen.

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