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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability1 Mark
MCQ
If $\text{f}(\text{x})=\frac{\sin^{-1}\text{x}}{\sqrt{1-\text{x}}^2},$ then $(1-\text{x})^2\text{f}\ ''(\text{x})-\text{xf}(\text{x})=$
  • ✓
    $1$
  • B
    $-1$
  • C
    $0$
  • D
    None of these
✓

Answer

Correct option: A.
$1$
Here,
$\text{f}(\text{x})=\frac{\sin^{-1}\text{x}}{\sqrt{1-\text{x}}^2}$
$\Rightarrow\sqrt{1-\text{x}}^2\text{f}(\text{x})=\sin^{-1}\text{x}$
Differentiating $\text{w.r.t. x},$ we get
$\sqrt{1-\text{x}^2}\text{f}'(\text{x})-\frac{\text{x f}{(\text{x})}}{\sqrt{1-\text{x}}^2}=\frac{1}{\sqrt{1-\text{x}}^2}$
$\Rightarrow(1-\text{x}^2)\text{f}\ '(\text{x})-\text{xf}(\text{x})=1$

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