Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(III), (B)-(I), (C)-(IV), (D)-(II)

The correct answer is Option (1) → (A)-(III), (B)-(I), (C)-(IV), (D)-(II)

$(A)\; y=\sin^{-1}x+\sin^{-1}\!\sqrt{1-x^{2}},\ |x|<1$

Differentiate:

$\frac{dy}{dx} =\frac{1}{\sqrt{1-x^{2}}}+\frac{-1}{\sqrt{1-x^{2}}} =0$

Hence derivative is $0$ → matches (III).

$(B)\; y=\sqrt{x+y},\ x+y>0,\ y\neq\frac12$

Square both sides: $y^{2}=x+y$

Differentiate: $2y\frac{dy}{dx}=1+\frac{dy}{dx}$

$\frac{dy}{dx}=\frac{1}{2y-1}$

Matches (I).

$(C)\; y=\log_{10}x,\ x>0$

$\frac{dy}{dx}=\frac{1}{x\ln 10}$

Matches (IV).

$(D)\; y=10^{x}$

$\frac{dy}{dx}=10^{x}\ln 10$

Matches (II).

Thus, the correct answer is: $(A\!-\!III),\ (B\!-\!I),\ (C\!-\!IV),\ (D\!-\!II)$.