The discussion focuses on differentiating the function y = tan^(-1)(x) with respect to x. The derivative is established as dy/dx = 1 / (1 + x^2), derived from the relationship x = tan(y) and the identity sec^2(y) = 1 + tan^2(y). This identity is confirmed through the fundamental trigonometric identity sin^2(y) + cos^2(y) = 1, which leads to the conclusion that sec^2(y) = 1 + x^2. The steps outlined provide a clear method for obtaining the derivative of the inverse tangent function.
PREREQUISITESStudents studying calculus, mathematics educators, and anyone seeking to deepen their understanding of differentiation involving inverse trigonometric functions.
It follows fromZedCar said:Homework Statement
Differentiate with respect to x
y = tan^(-1) x
Homework Equations
The Attempt at a Solution
ANSWER in book:
y = tan^(-1) x
x = tan y
dx/dy = sec^(2) y = (1 + x^2)
dy/dx = 1 / (1 + x^2)
How is it known that sec^(2) y = (1 + x^2)
Does it follow from an identity?