Find dy/dx if y = tan⁻¹((1 + x)/(1 − x)), −1 < x < 1.

How to lay out the answer. Follow the frame, then write it in your own words.

1 · Write what is given and what is to be found. Given: y = tan⁻¹((1 + x)/(1 − x)), −1 < x < 1. To find: dy/dx.
2 · Choose a substitution. Put x = tan θ. Since −1 < x < 1, −π/4 < θ < π/4.
3 · Simplify the expression. (1 + tan θ)/(1 − tan θ) = tan(π/4 + θ), using tan(A + B) with tan(π/4) = 1.
4 · Remove tan⁻¹. 0 < π/4 + θ < π/2, which is in the principal value branch. So y = π/4 + θ = π/4 + tan⁻¹x.
5 · Differentiate. dy/dx = 0 + 1/(1 + x²).
6 · Check and box the answer. Differentiating directly with the chain rule also gives 1/(1 + x²). Answer: dy/dx = 1/(1 + x²).

Scoring tip: Always state the range of θ. Without it, writing tan⁻¹(tan A) = A loses a mark.