Derivatives And Chain Rule Doubt.

Question

For some function,

$x = f(t)$

$y = g(t)$

If we wish to calculate second order derivative of y with respect to x would it be right to approach in the following manner:

$\frac{d^2y}{dx^2} = \frac{d^2y}{dt^2}.\frac{dt^2}{dx^2}$ ?

Answer 1

No.

\begin{eqnarray*} \frac{\mathrm d^2y}{\mathrm dx^2} &=& \frac{\mathrm d}{\mathrm dx}\frac{\mathrm dy}{\mathrm dx} \\ &=& \frac{\mathrm d}{\mathrm dx}\left(\frac{\mathrm dy}{\mathrm dt}\cdot\frac{\mathrm dt}{\mathrm dx}\right) \\ &=& \left(\frac{\mathrm d}{\mathrm dx}\frac{\mathrm dy}{\mathrm dt}\right)\frac{\mathrm dt}{\mathrm dx}+\frac{\mathrm dy}{\mathrm dt}\cdot\left(\frac{\mathrm d}{\mathrm dx}\frac{\mathrm dt}{\mathrm dx}\right) \\ &=& \frac{\mathrm d^2y}{\mathrm dt^2}\cdot\left(\frac{\mathrm dt}{\mathrm dx}\right)^2+\frac{\mathrm dy}{\mathrm dt}\cdot\frac{\mathrm d^2t}{\mathrm dx^2}\;. \end{eqnarray*}