Differentiation Question 359
Question: If $ y={e^{{{\tan }^{-1}}x}} $ , then $ (1+x^{2})\frac{d^{2}y}{dx^{2}}= $
Options:
A) $ (1-2x)\frac{dy}{dx} $
B) $ -2x\frac{dy}{dx} $
C) $ -x\frac{dy}{dx} $
D) 0
Show Answer
Answer:
Correct Answer: A
Solution:
$ y={e^{{{\tan }^{-1}}x}}\Rightarrow \frac{dy}{dx}=\frac{{e^{{{\tan }^{-1}}x}}}{1+x^{2}} $
$ \Rightarrow \frac{d^{2}y}{dx^{2}}=\frac{(1+x^{2}).\frac{{e^{{{\tan }^{-1}}x}}}{(1+x^{2})}-{e^{{{\tan }^{-1}}x}}(2x)}{{{(1+x^{2})}^{2}}} $
$ \Rightarrow \frac{d^{2}y}{dx^{2}}=\frac{(1-2x){e^{{{\tan }^{-1}}x}}}{{{(1+x^{2})}^{2}}} $
$ \Rightarrow \frac{d^{2}y}{dx^{2}}(1+x^{2})=(1-2x)\frac{dy}{dx} $ .
BETA
