Derivatives of Inverse Trigonometric Functions

A ship tracks an approaching aircraft by measuring the slant distance and the angle of elevation. From these measurements, the horizontal and vertical components of the distance can be found using trigonometric relationships

To analyze the change in angle mathematically, angle theta is mapped as y, and the Altitude over distance ratio is mapped as x, giving the relation y equals arcsin x.

The measure of how instantaneous changes in x affect y is the angle sensitivity, and this requires implicit differentiation.

The sine of both sides of the equation provides an equivalent trigonometric form, expressed as sine y equals x.

This equation is then differentiated implicitly with respect to x to find the derivative.

By rearranging and applying the Pythagorean identity, the cosine function can be written in terms of sine. Since the sine of y equals x, this can be substituted back to get the derivative of arcsin of x.

When the aircraft is overhead, a tiny change in x produces a huge change in the angle, resulting in tracking sensitivity explosion and subsequent tracking failure. Similarly, the derivatives of the remaining inverse trigonometric functions can also be found.