Answer

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Hint: The given expression should be derived with respect to $’y’$ and not $’x’$.

The given expression is

\[y=x+{{e}^{x}}\]

Now we will find the first order derivative of the given expression, so we will differentiate the given

expression with respect to $'y'$, we get

\[\dfrac{d}{dy}(y)=\dfrac{d}{dy}\left( x+{{e}^{x}} \right)\]

Now we will apply the the sum rule of differentiation, i.e., differentiation of sum of two functions is same as the sum of individual differentiation of the functions, i.e.,

$\dfrac{d}{dx}(u+v)=\dfrac{d}{x}(u)+\dfrac{d}{x}(v)$. Applying this formula in the above equation, we get

\[\dfrac{d}{dy}(y)=\dfrac{d}{dy}\left( x \right)+\dfrac{d}{dy}\left( {{e}^{x}} \right)\]

We know differentiation of an exponential function is, $\dfrac{d}{dx}\left( {{e}^{u}}

\right)={{e}^{u}}.\dfrac{d}{dx}(u)$, so the above equation becomes,

\[\dfrac{d}{dy}(y)=\dfrac{dx}{dy}+{{e}^{x}}\dfrac{d}{dy}\left( x \right)\]

We know differentiation, \[\dfrac{d}{dx}\left( x \right)=1\] , so the above equation becomes,

\[1=\dfrac{dx}{dy}\left( 1+{{e}^{x}} \right)\]

\[\dfrac{dx}{dy}=\dfrac{1}{1+{{e}^{x}}}={{\left( 1+{{e}^{x}} \right)}^{-1}}........(i)\]

Now we will find the second order derivative. For that we will again differentiate the above

expression with respect to $'y'$ , we get

\[\dfrac{d}{dy}\left( \dfrac{dx}{dy} \right)=\dfrac{d}{dy}{{\left( 1+{{e}^{x}} \right)}^{-1}}\]

Now we know $\dfrac{d}{dx}({{u}^{n}})=n{{u}^{n-1}}\dfrac{d}{dx}(u)$ , applying this formula, the above equation becomes,

\[\dfrac{{{d}^{2}}x}{d{{y}^{2}}}=(-1){{\left( 1+{{e}^{x}} \right)}^{-1-1}}\dfrac{d}{dy}\left( 1+{{e}^{x}}

\right)\]

Now we will apply the the sum rule of differentiation, i.e., differentiation of sum of two functions is same as the sum of individual differentiation of the functions, i.e.,

$\dfrac{d}{dx}(u+v)=\dfrac{d}{x}(u)+\dfrac{d}{x}(v)$ . Applying this formula in the above equation, we get

\[\dfrac{{{d}^{2}}x}{d{{y}^{2}}}=(-1){{\left( 1+{{e}^{x}} \right)}^{-2}}\left[ \dfrac{d}{dy}\left( 1

\right)+\dfrac{d}{dy}\left( {{e}^{x}} \right) \right]\]

We know differentiation of an exponential function is, $\dfrac{d}{dx}\left( {{e}^{u}}

\right)={{e}^{u}}.\dfrac{d}{dx}(u)$, so the above equation becomes,

\[\dfrac{{{d}^{2}}x}{d{{y}^{2}}}=(-1){{\left( 1+{{e}^{x}} \right)}^{-2}}\left[ \dfrac{d}{dy}\left( 1

\right)+{{e}^{x}}\dfrac{d}{dy}\left( x \right) \right]\]

We know differentiation of a constant is always a zero, so above equation can be written as,

\[\dfrac{{{d}^{2}}x}{d{{y}^{2}}}=(-1){{\left( 1+{{e}^{x}} \right)}^{-2}}\left[ 0+{{e}^{x}}\dfrac{dx}{dy}

\right]\]

Now substituting from equation (i), the above equation becomes,

\[\begin{align}

& \dfrac{{{d}^{2}}x}{d{{y}^{2}}}=(-1){{\left( 1+{{e}^{x}} \right)}^{-2}}\left[ {{e}^{x}}{{\left( 1+{{e}^{x}}

\right)}^{-1}} \right] \\

& \Rightarrow \dfrac{{{d}^{2}}x}{d{{y}^{2}}}=\dfrac{-{{e}^{x}}}{{{\left( 1+{{e}^{x}} \right)}^{2+1}}} \\

& \Rightarrow \dfrac{{{d}^{2}}x}{d{{y}^{2}}}=\dfrac{-{{e}^{x}}}{{{\left( 1+{{e}^{x}} \right)}^{3}}} \\

\end{align}\]

Hence the correct answer is option (b).

Note: We know derivative of \[{{e}^{x}}\] is \[{{e}^{x}}\], but this is with respect to $x$, if the derivative is with respect to any other variable, then we cannot assume this, we should use the formula$\dfrac{d}{dx}\left( {{e}^{u}} \right)={{e}^{u}}.\dfrac{d}{dx}(u)$. Whenever derivating we should pay attention to the variable it is being derived with respect to. As in this problem see the simple expression student will derive with respect to $'x'$ instead of $'y'$ and will get an incorrect answer.

The given expression is

\[y=x+{{e}^{x}}\]

Now we will find the first order derivative of the given expression, so we will differentiate the given

expression with respect to $'y'$, we get

\[\dfrac{d}{dy}(y)=\dfrac{d}{dy}\left( x+{{e}^{x}} \right)\]

Now we will apply the the sum rule of differentiation, i.e., differentiation of sum of two functions is same as the sum of individual differentiation of the functions, i.e.,

$\dfrac{d}{dx}(u+v)=\dfrac{d}{x}(u)+\dfrac{d}{x}(v)$. Applying this formula in the above equation, we get

\[\dfrac{d}{dy}(y)=\dfrac{d}{dy}\left( x \right)+\dfrac{d}{dy}\left( {{e}^{x}} \right)\]

We know differentiation of an exponential function is, $\dfrac{d}{dx}\left( {{e}^{u}}

\right)={{e}^{u}}.\dfrac{d}{dx}(u)$, so the above equation becomes,

\[\dfrac{d}{dy}(y)=\dfrac{dx}{dy}+{{e}^{x}}\dfrac{d}{dy}\left( x \right)\]

We know differentiation, \[\dfrac{d}{dx}\left( x \right)=1\] , so the above equation becomes,

\[1=\dfrac{dx}{dy}\left( 1+{{e}^{x}} \right)\]

\[\dfrac{dx}{dy}=\dfrac{1}{1+{{e}^{x}}}={{\left( 1+{{e}^{x}} \right)}^{-1}}........(i)\]

Now we will find the second order derivative. For that we will again differentiate the above

expression with respect to $'y'$ , we get

\[\dfrac{d}{dy}\left( \dfrac{dx}{dy} \right)=\dfrac{d}{dy}{{\left( 1+{{e}^{x}} \right)}^{-1}}\]

Now we know $\dfrac{d}{dx}({{u}^{n}})=n{{u}^{n-1}}\dfrac{d}{dx}(u)$ , applying this formula, the above equation becomes,

\[\dfrac{{{d}^{2}}x}{d{{y}^{2}}}=(-1){{\left( 1+{{e}^{x}} \right)}^{-1-1}}\dfrac{d}{dy}\left( 1+{{e}^{x}}

\right)\]

Now we will apply the the sum rule of differentiation, i.e., differentiation of sum of two functions is same as the sum of individual differentiation of the functions, i.e.,

$\dfrac{d}{dx}(u+v)=\dfrac{d}{x}(u)+\dfrac{d}{x}(v)$ . Applying this formula in the above equation, we get

\[\dfrac{{{d}^{2}}x}{d{{y}^{2}}}=(-1){{\left( 1+{{e}^{x}} \right)}^{-2}}\left[ \dfrac{d}{dy}\left( 1

\right)+\dfrac{d}{dy}\left( {{e}^{x}} \right) \right]\]

We know differentiation of an exponential function is, $\dfrac{d}{dx}\left( {{e}^{u}}

\right)={{e}^{u}}.\dfrac{d}{dx}(u)$, so the above equation becomes,

\[\dfrac{{{d}^{2}}x}{d{{y}^{2}}}=(-1){{\left( 1+{{e}^{x}} \right)}^{-2}}\left[ \dfrac{d}{dy}\left( 1

\right)+{{e}^{x}}\dfrac{d}{dy}\left( x \right) \right]\]

We know differentiation of a constant is always a zero, so above equation can be written as,

\[\dfrac{{{d}^{2}}x}{d{{y}^{2}}}=(-1){{\left( 1+{{e}^{x}} \right)}^{-2}}\left[ 0+{{e}^{x}}\dfrac{dx}{dy}

\right]\]

Now substituting from equation (i), the above equation becomes,

\[\begin{align}

& \dfrac{{{d}^{2}}x}{d{{y}^{2}}}=(-1){{\left( 1+{{e}^{x}} \right)}^{-2}}\left[ {{e}^{x}}{{\left( 1+{{e}^{x}}

\right)}^{-1}} \right] \\

& \Rightarrow \dfrac{{{d}^{2}}x}{d{{y}^{2}}}=\dfrac{-{{e}^{x}}}{{{\left( 1+{{e}^{x}} \right)}^{2+1}}} \\

& \Rightarrow \dfrac{{{d}^{2}}x}{d{{y}^{2}}}=\dfrac{-{{e}^{x}}}{{{\left( 1+{{e}^{x}} \right)}^{3}}} \\

\end{align}\]

Hence the correct answer is option (b).

Note: We know derivative of \[{{e}^{x}}\] is \[{{e}^{x}}\], but this is with respect to $x$, if the derivative is with respect to any other variable, then we cannot assume this, we should use the formula$\dfrac{d}{dx}\left( {{e}^{u}} \right)={{e}^{u}}.\dfrac{d}{dx}(u)$. Whenever derivating we should pay attention to the variable it is being derived with respect to. As in this problem see the simple expression student will derive with respect to $'x'$ instead of $'y'$ and will get an incorrect answer.

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