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Some Class 12 differentiation questions look ugly: y = sin⁻¹(2x/(1 + x²)), or y = tan⁻¹((3x − x³)/(1 − 3x²)). Differentiating them directly with the chain rule works, but it is long and easy to get wrong. A trigonometric substitution turns each one into a multiple of tan⁻¹x or sin⁻¹x in a few lines. This post shows the method, the identities it relies on, and the domain trap that catches many students.
Where is this in the syllabus?
The CBSE Mathematics curriculum for 2026-27 lists, for Class 12, "derivatives of inverse trigonometric functions like sin⁻¹x, cos⁻¹x and tan⁻¹x". NCERT Class 12 Chapter 5, Continuity and Differentiability sets this type in Exercise 5.3, Questions 9 to 15, each with an interval such as −1/√3 < x < 1/√3 or 0 < x < 1. The JEE (Main) 2026 syllabus includes differentiation of inverse trigonometric and composite functions. Check your exam year's documents. For principal values, read our post on inverse trigonometric functions first.
What is the method?
Look at the expression inside the inverse function and match it to a double- or triple-angle formula.
Substitute: x = tan θ for 1 + x² or 1 − x², x = sin θ or cos θ for √(1 − x²).
Simplify the inside to a single trigonometric function of 2θ or 3θ.
Use the given interval to decide whether sin⁻¹(sin 2θ) really equals 2θ.
Write the answer in terms of x and differentiate.
The identities you need: sin 2θ = 2 tan θ/(1 + tan²θ), cos 2θ = (1 − tan²θ)/(1 + tan²θ), tan 2θ = 2 tan θ/(1 − tan²θ), and tan 3θ = (3 tan θ − tan³θ)/(1 − 3 tan²θ).
Worked example 1: y = tan⁻¹((3x − x³)/(1 − 3x²)), −1/√3 < x < 1/√3
Put x = tan θ. The inside becomes tan 3θ.
For −1/√3 < x < 1/√3, θ lies between −π/6 and π/6, so 3θ lies between −π/2 and π/2, the principal range of tan⁻¹.
So y = 3θ = 3 tan⁻¹x, and dy/dx = 3/(1 + x²).
Worked example 2: y = cos⁻¹((1 − x²)/(1 + x²)), 0 < x < 1
Put x = tan θ. The inside becomes cos 2θ.
For 0 < x < 1, θ lies between 0 and π/4, so 2θ lies between 0 and π/2, inside the principal range [0, π] of cos⁻¹.
So y = 2 tan⁻¹x and dy/dx = 2/(1 + x²).
Worked example 3: the domain trap
Let y = sin⁻¹(2x/(1 + x²)). Put x = tan θ, so the inside is sin 2θ.
If |x| ≤ 1, then |θ| ≤ π/4 and |2θ| ≤ π/2. So y = 2θ = 2 tan⁻¹x, and dy/dx = 2/(1 + x²).
If x > 1, then π/4 < θ < π/2 and 2θ lies between π/2 and π, outside the principal range of sin⁻¹. Here sin⁻¹(sin 2θ) = π − 2θ, so y = π − 2 tan⁻¹x and dy/dx = −2/(1 + x²).
Check with the chain rule. With u = 2x/(1 + x²), u′ = 2(1 − x²)/(1 + x²)² and √(1 − u²) = |1 − x²|/(1 + x²). So dy/dx = 2(1 − x²)/((1 + x²)|1 − x²|), which is 2/(1 + x²) when |x| < 1 and −2/(1 + x²) when |x| > 1. Both methods agree, and the absolute value is exactly where the sign change comes from. At x = 1 the function is not differentiable.
A sin-substitution example
NCERT's Class 12 Chapter 2 on inverse trigonometric functions shows in Example 3 that sin⁻¹(2x√(1 − x²)) = 2 sin⁻¹x for −1/√2 ≤ x ≤ 1/√2, using x = sin θ. On that interval, dy/dx = 2/√(1 − x²). For 1/√2 ≤ x ≤ 1 the same example gives 2 cos⁻¹x, so the derivative becomes −2/√(1 − x²).
Common mistakes and how to avoid them
Ignoring the interval. Always find where 2θ or 3θ lies.
Choosing the wrong substitution. 1 + x² suggests tan θ; √(1 − x²) suggests sin θ or cos θ.
Forgetting to convert back to x before differentiating.
Assuming the derivative of cos⁻¹(...) carries a minus sign after simplification. Here y = 2 tan⁻¹x, so it does not.
Why check by the chain rule as well?
Rittle-Johnson and Star (2007) found that 70 seventh-graders who compared two solution methods side by side made greater gains in procedural knowledge and flexibility than those who studied the methods one at a time. Doing one problem by substitution and by the chain rule shows you why the interval matters, not just that it does.
A practice routine
Work through NCERT Exercise 5.3, Questions 9 to 15, writing the range of θ for each.
For two of them, check your answer with the chain rule.
Make up one problem with x > 1 and find the derivative there.
Revisit two problems a week later without notes.
If a concept such as principal value branches is still shaky, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, and you can read the idea before returning to the exercise.
Next step: differentiate y = sin⁻¹((1 − x²)/(1 + x²)) for 0 < x < 1 using x = tan θ, then check it with the chain rule.
