Match List-I with List-II: List-I List-II (A) The derivative of log_e…

The correct answer is Option (2) → (A)-(I), (B)-(III), (C)-(II), (D)-(IV)

List-I	List-II
(A) The derivative of $\log_ex$ with respect to $\left(\frac{1}{x}\right)$ at $x=5$ is	(I) -5
(B) If $x^3+x^2y+xy^2=21x$, then $\frac{dy}{dx}$ at (1, 1) is	(III) 5
(C) If $f(x)=x^3\log_e\frac{1}{x}$, then $f'(1)+f''(1)$ is	(II) -6
(D) If $y=f(x^2)$ and $f'(x)=e^{\sqrt{x}}$, then $\frac{dy}{dx}$ at $x=0$ is	(IV) 0

$(A)\;\frac{d}{d(1/x)}(\log_ex)=\frac{\frac{d}{dx}(\log_ex)}{\frac{d}{dx}(1/x)}=\frac{\frac{1}{x}}{-\frac{1}{x^2}}=-x.$

$\text{At }x=5,\;=-5\Rightarrow(I).$

$(B)\;x^3+x^2y+xy^2=21x.$

$3x^2+2xy+x^2\frac{dy}{dx}+y^2+2xy\frac{dy}{dx}=21.$

$(x^2+2xy)\frac{dy}{dx}=21-3x^2-2xy-y^2.$

$\text{At }(1,1):\;(1+2)\frac{dy}{dx}=21-3-2-1=15.$

$3\frac{dy}{dx}=15\Rightarrow\frac{dy}{dx}=5\Rightarrow(III).$

$(C)\;f(x)=x^3\log_e\frac{1}{x}.$

$f'(x)=3x^2\log_e\frac{1}{x}-x^2.$

$f''(x)=6x\log_e\frac{1}{x}-5x.$

$f'(1)=0-1=-1,\;f''(1)=0-5=-5.$

$f'(1)+f''(1)=-6\Rightarrow(II).$

$(D)\;y=f(x^2),\;\frac{dy}{dx}=2xf'(x^2).$

$\text{At }x=0,\;\frac{dy}{dx}=0\Rightarrow(IV).$

$(A)\rightarrow(I),\;(B)\rightarrow(III),\;(C)\rightarrow(II),\;(D)\rightarrow(IV).$