Question

If \[x=4t\] and \[y=2{{t}^{2}}\], then \[\dfrac{{{d}^{2}}y}{d{{x}^{2}}}\]at \[x=\dfrac{1}{2}\] is
       \[\begin{align}
  & \left( \text{a} \right)\text{ }\dfrac{1}{2} \\
 & \left( b \right)\text{ }\dfrac{1}{4} \\
 & \left( c \right)\text{ 2} \\
 & \left( d \right)\text{ 4} \\
\end{align}\]

Answer 1

- Hint:Consider ‘x’ and ‘y’ separately, then differentiate them with respect to ‘t’ separately. Then apply the parametric form of derivation, i.e., \[\dfrac{dy}{dx}=\dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}\]. Then substitute the given values and find out the value of ‘t’. Later differentiate again to get a second order derivative and solve further.

Complete step-by-step solution -
 As per the given information, \[x=4t\], \[y=2{{t}^{2}}\].
For this first let us find \[\dfrac{dy}{dt}\And \dfrac{dx}{dt}.\]
First we will derive $'y'$ with respect to $'t'$, we get
\[\dfrac{dy}{dt}=\dfrac{d}{dt}(2{{t}^{2}})\]
Differentiating on both the sides, we get
\[\dfrac{dy}{dt}=2.2t=4t............(i)\]
Now we will derive $'x'$ with respect to $'t'$, we get
\[\dfrac{dx}{dt}=\dfrac{d}{dt}(4t)\]
By differentiating on both sides, we get
\[\dfrac{dx}{dt}=4(1)=4..............(ii)\]
Now dividing equation (i) by equation (ii), we get
\[\dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}=\dfrac{4t}{4}\]
Cancelling the like terms, we get
\[\Rightarrow \dfrac{dy}{dx}=t.......(iii)\]
Now given \[x=4t\], so
\[\Rightarrow t=\dfrac{x}{4}\]
Substitute this in equation (iii), we get
\[\dfrac{dy}{dx}=\dfrac{x}{4}\]
Now, we will again differentiate both sides with respect to ‘x’, we get
\[\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=\dfrac{d}{dx}\left( \dfrac{x}{4} \right)\]
Taking out the constant term, we get
\[\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=\dfrac{1}{4}.\dfrac{d}{dx}(x)=\dfrac{1}{4}(1)\]
\[\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=\dfrac{1}{4}\]
And as the second derivative is a constant, which is independent of x-value, so it will be the same value at any point, i.e.,
\[{{\left( \dfrac{{{d}^{2}}y}{d{{x}^{2}}} \right)}_{x=\dfrac{1}{2}}}=\dfrac{1}{4}\]
Hence the correct answer is option (b).

Note: The given equations are of parametric form. Here we can also find the value of $'t'$from the given value of $'x'$, i.e., \[t=\dfrac{x}{4}\].
Now substituting the value of $'t'$in the given value of $'y'$, i.e., \[y=2{{\left( \dfrac{x}{4} \right)}^{2}}\]. Here we can observe that $'y'$is independent of $'t'$. Now we can differentiate $'y'$directly with respect to $'x'$ and find the required values.