Ohio Assessments for Educators (OAE) Mathematics Practice Exam

The derivative of the inverse tangent function, tan⁻¹(u), is derived from the fundamental principles of calculus. To find the rate of change of the inverse tangent function concerning its argument u, the formula for the derivative is well-established. The derivative of tan⁻¹(u) is given by the expression:

1/(1 + u²).

This formula reflects how the slope of the tangent line to the graph of the inverse tangent function varies based on the value of u. The additional factor of du/dx indicates that we are applying the chain rule, which accounts for the dependence on x through u. Therefore, the complete derivative involves multiplying the fundamental derivative 1/(1 + u²) by the derivative of u with respect to x.

This relationship shows that as u increases or decreases, the rate of change of tan⁻¹(u) varies, influenced by the term (1 + u²) which ensures the output responds appropriately to changes in u, remaining well-defined for all real numbers. This is why the expression 1/(1 + u²) du/dx is the correct representation of the derivative of tan⁻¹(u).