Final answer:
Absolute value can often be dropped from the derivatives of inverse trigonometric functions due to the principal domain and range, which ensure the arguments remain in a range where the absolute value is not necessary.
Step-by-step explanation:
When working with inverse trigonometric functions in calculus, the derivatives sometimes involve absolute value signs due to the properties of those functions. For instance, the derivative of arcsin has the form 1 / √(1 - x2), but for reasons tied to domain and range, we often do not retain the absolute value in practical application.
The justification for 'dropping the absolute value' in the context of derivatives of inverse trigonometric functions lies primarily in assuming that the argument, or the variable within the function, adheres to the principal domain. For example, when dealing with arcsin(x) or arccos(x), the principal domain is [-1, 1], which guarantees that the values within the square root are non-negative, thus rendering the absolute value unnecessary.
Another aspect is the principal range of these functions, which determines the sign of the square root in the derivative. For instance, since the range of arcsin(x) is [-π/2, π/2], all values that x can be are such that the square root of (1 - x2) will be non-negative. This also applies to arccos(x) with its range of [0, π].
In summary, the domains and ranges of the inverse trigonometric functions imply specific restrictions on the input values, allowing us to simplify the derivative expressions and ensuring that the functions behave predictably within those boundaries. Dropping the absolute value doesn't change the correctness of the derivative, but rather reflects the inherent constraints of the function being differentiated.