Maths · Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions
If \) then
If \( y=\sin ^{-1} \frac{1}{2}(\sqrt{1+x}+\sqrt{1-x}) \) then \( y^{\prime}= \)
- A. \( \frac{1}{2 \sqrt{1-x^{2}}} \)
- B. \( \frac{-1}{2 \sqrt{1-x^{2}}} \)
- C. \( \frac{1}{2 \sqrt{1+x^{2}}} \)
- D. \( \frac{-1}{2 \sqrt{1+x^{2}}} \)
Step-by-step solution
Let y = sin^{-1}( (√(1+x)+√(1-x))/2 ). For x in [0,1], substitute x = cos(2θ) to simplify: y = π/2 - (1/2)arcsin x. Differentiating gives y' = -1/(2√(1-x^2)). For x in [-1,0], the derivative is positive, but the problem likely assumes the principal branch where x∈[0,1], yielding a negative derivative. Among options, only B matches this form.