If y = x + e^x , then d^2 x d y^2 = - Vidyadip

MCQ

If $y = \sin x + {e^x},$ then ${{{d^2}x} \over {d{y^2}}} = $

A
${( - \sin x + {e^x})^{ - 1}}$
B
${{\sin x - {e^x}} \over {{{(\cos x + {e^x})}^2}}}$
✓
${{\sin x - {e^x}} \over {{{(\cos x + {e^x})}^3}}}$
D
${{\sin x + {e^x}} \over {{{(\cos x + {e^x})}^3}}}$

✓

Answer

Correct option: C.

${{\sin x - {e^x}} \over {{{(\cos x + {e^x})}^3}}}$

c
(c) $y = \sin x + {e^x}$==> $\frac{{dy}}{{dx}} = \cos x + {e^x}$

==> $\frac{{dx}}{{dy}} = {(\cos x + {e^x})^{ - 1}}$ …..$(i)$

Again, $\frac{{{d^2}x}}{{d{y^2}}} = - {(\cos x + {e^x})^{ - 2}}( - \sin x + {e^x})\frac{{dx}}{{dy}}$.

Substituting the value of $\frac{{dx}}{{dy}}$ from $(i),$

$\frac{{{d^2}x}}{{d{y^2}}} = \frac{{(\sin x - {e^x})}}{{{{(\cos x + {e^x})}^2}}}\,{(\cos x + {e^x})^{ - 1}}$

$ = \frac{{\sin x - {e^x}}}{{{{(\cos x + {e^x})}^3}}}$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $z = 1 + ai$ be a complex number, $a > 0$, such that $z^3$ is areal number. Then the sum $1 + z + z^2 + .... + z^{11}$ is equal to

Let $P$ is a point on hyperbola $x^2 -y^2 = 4$ , which is at minimum distance from $(0,-1)$ then distance of $P$ from $x-$ axis is

The number of integral terms in the expansion of $(7^{1/3} + 11^{1/9})^{6561}$ is :-

A coin is tossed twice. The probability of getting head both the times is

The value of expression $\left( {\cos \frac{\pi }{2} + i\sin \frac{\pi }{2}} \right)$ $\,\left( {\cos \frac{\pi }{{{2^2}}} + i\sin \frac{\pi }{{{2^2}}}} \right)$........to $\infty $ is

If the angles of a quadrilateral are in $A.P.$ whose common difference is ${10^o}$, then the angles of the quadrilateral are

If $P(A) = \frac{1}{2},\,\,P(B) = \frac{1}{3}$ and $P(A \cap B) = \frac{7}{{12}},$ then the value of $P\,(A' \cap B')$ is

Let $f: R \rightarrow R$ be a function defined as $f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x], a \in R$, where [t] is the greatest integer less than or equal to $t$. If $\lim _{x \rightarrow-1} f(x)$ exists, then the value of $\int_{0}^{4} f(x) d x$ is equal to.

The focus of the parabola $y = 2{x^2} + x$ is

If $f(x)\, = \,\left\{ {\begin{array}{*{20}{c}}{\frac{{\sqrt {1 + kx} - \sqrt {1 - kx} }}{x}}&{{\rm{,for}} - 1 \le x < 0}\\{2{x^2} + 3x - 2}&{{\rm{,}}\,{\rm{for\,\, }}\,0 \le \,x \le 1}\end{array}} \right.$, is continuous at $x = 0$, then $k = $