MCQ
If $y = {e^{{{\tan }^{ - 1}}x}}$, then $(1 + {x^2}){{{d^2}y} \over {d{x^2}}} = $
- ✓$(1 - 2x){{dy} \over {dx}}$
- B$ - 2x{{dy} \over {dx}}$
- C$ - x{{dy} \over {dx}}$
- D$0$
✓
Answer
Correct option: A.
$(1 - 2x){{dy} \over {dx}}$
a
(a) $y = {e^{{{\tan }^{ - 1}}x}} \Rightarrow \frac{{dy}}{{dx}} = \frac{{{e^{{{\tan }^{ - 1}}x}}}}{{1 + {x^2}}}$
(a) $y = {e^{{{\tan }^{ - 1}}x}} \Rightarrow \frac{{dy}}{{dx}} = \frac{{{e^{{{\tan }^{ - 1}}x}}}}{{1 + {x^2}}}$
$ \Rightarrow \frac{{{d^2}y}}{{d{x^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}}{{d{x^2}}} = \frac{{(1 - 2x){e^{{{\tan }^{ - 1}}x}}}}{{{{(1 + {x^2})}^2}}}$
$ \Rightarrow \frac{{{d^2}y}}{{d{x^2}}}(1 + {x^2}) = (1 - 2x)\frac{{dy}}{{dx}}$.
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