Limits, Continuity & Differentiability
Differentiation
Grade 12
Question:
<p>The derivative of \(\tan^{-1}\left(\dfrac{\sin x - \cos x}{\sin x + \cos x}\right)\), with respect to \(\dfrac{x}{2}\), where \(x \in \left(0, \dfrac{\pi}{2}\right)\) is:</p>
<p>1</p>
<p>\(\dfrac{2}{3}\)</p>
<p>\(\dfrac{1}{2}\)</p>
<p>2</p>
Step-by-Step Solution
Key Concept: Recognize that sin x - cos x = √2 sin(x - π/4) and sin x + cos x = √2 sin(x + π/4), allowing you to simplify the argument of tan⁻¹ using the tangent subtraction formula. This transforms the problem into finding d/d(x/2) of a simple expression.
<p><strong>Step 1:</strong> Simplify the argument of tan⁻¹</p><p>Let y = tan⁻¹((sin x - cos x)/(sin x + cos x))</p><p>Divide numerator and denominator by cos x:</p><p>y = tan⁻¹((tan x - 1)/(tan x + 1))</p><p><strong>Step 2:</strong> Apply tangent subtraction formula</p><p>Note that tan(A - B) = (tan A - tan B)/(1 + tan A tan B)</p><p>With A = x and B = π/4: tan(x - π/4) = (tan x - 1)/(1 + tan x)</p><p>Therefore: y = tan⁻¹(tan(x - π/4)) = x - π/4 (since x ∈ (0, π/2), we have x - π/4 ∈ (-π/4, π/4))</p><p><strong>Step 3:</strong> Differentiate with respect to x/2</p><p>dy/d(x/2) = dy/dx × dx/d(x/2)</p><p>dy/dx = d/dx(x - π/4) = 1</p><p>dx/d(x/2) = 2</p><p>∴ dy/d(x/2) = 1 × 2 = <strong>2</strong></p>
Correct Answer: D