Question

If \[x = a{t^2},y = 2at\] then find the value of \[\dfrac{{{d^2}y}}{{d{x^2}}}\] .
A.\[ - \dfrac{1}{{{t^2}}}\]
B. \[ - \dfrac{1}{{{t^3}}}\]
C. \[ - \dfrac{1}{{2a{t^3}}}\]
D. \[\dfrac{1}{{2a{t^3}}}\]

Answer 1

Hint:First differentiate the given equation of x with respect to t, then differentiate the given equation of y with respect to t. After that divide \[\dfrac{{dy}}{{dt}}\] by \[\dfrac{{dx}}{{dt}}\] to obtain the first order derivative of y with respect to x. Then again differentiate \[\dfrac{{dy}}{{dx}}\] with respect to x and calculate to obtain the required result.

Formula used
\[\dfrac{d}{{dx}}(af(x)) = a\dfrac{d}{{dx}}f(x)\]
\[\dfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}\]

Complete step by step solution
It is given that \[x = a{t^2},y = 2at\].
Differentiate x with respect to t.
Therefore,
\[\dfrac{{dx}}{{dt}} = 2at\]
Differentiate y with respect to t.
Therefore,
\[\dfrac{{dy}}{{dt}} = 2a\]
Divide \[\dfrac{{dy}}{{dt}}\] by \[\dfrac{{dx}}{{dt}}\]to obtain the first order derivative of y with respect to x.
Hence,
\[\dfrac{{\dfrac{{dy}}{{dt}}}}{{\dfrac{{dx}}{{dt}}}} = \dfrac{{2a}}{{2at}}\]
\[\dfrac{{dy}}{{dx}} = \dfrac{1}{t}\]
Now, differentiate with respect to x to obtain the required answer.
\[\dfrac{d}{{dx}}\left( {\dfrac{{dy}}{{dx}}} \right) = \dfrac{d}{{dx}}\left( {\dfrac{1}{t}} \right)\]
\[\dfrac{{{d^2}y}}{{d{x^2}}} = \dfrac{d}{{dt}}\left( {\dfrac{1}{t}} \right).\dfrac{{dt}}{{dx}}\]
\[ = - \dfrac{1}{{{t^2}}}.\dfrac{1}{{\dfrac{{dx}}{{dt}}}}\]
\[ = - \dfrac{1}{{{t^2}}}.\dfrac{1}{{2at}}\]
\[ = - \dfrac{1}{{2a{t^3}}}\]

The correct option is C.


Note Students often forget to multiply the term \[\dfrac{{dt}}{{dx}}\] in the step \[\dfrac{{{d^2}y}}{{d{x^2}}} = \dfrac{d}{{dt}}\left( {\dfrac{1}{t}} \right).\dfrac{{dt}}{{dx}}\] and write the correct option as A, that is not correct. We cannot differentiate t with respect to x, so as we are changing dx to dt so we have to multiply the term by \[\dfrac{{dt}}{{dx}}\] to keep the equation as it is.