Limits, Continuity & Differentiability
Differentiation of Inverse Trigonometric Functions
Grade 12
<p>If \( y = \sec(\tan^{-1} x) \), then \( \dfrac{dy}{dx} \) at \( x = 1 \) is equal to:</p>
Step-by-Step Solution
Key Concept: Use the right triangle interpretation of inverse trigonometric functions: if θ = tan⁻¹(x), then tan(θ) = x, which means we can construct a right triangle with opposite = x, adjacent = 1, and hypotenuse = √(1+x²). Then sec(θ) = hypotenuse/adjacent = √(1+x²).
<p><strong>Step 1:</strong> Simplify y = sec(tan⁻¹ x) using the right triangle method.</p><p>Let θ = tan⁻¹(x), so tan(θ) = x</p><p>In a right triangle: opposite = x, adjacent = 1, hypotenuse = √(1+x²)</p><p>Therefore: sec(θ) = √(1+x²)</p><p><strong>Step 2:</strong> So y = √(1+x²) = (1+x²)^(1/2)</p><p><strong>Step 3:</strong> Differentiate using power rule and chain rule:</p><p>dy/dx = (1/2)(1+x²)^(-1/2) · 2x = x/√(1+x²)</p><p><strong>Step 4:</strong> Evaluate at x = 1:</p><p>dy/dx|ₓ₌₁ = 1/√(1+1) = 1/√2 = √2/2</p><p>∴ Answer: D</p>
Correct Answer: D