Hi, hoping that someone here can help shed some light on this for me.

You are given that $\displaystyle x = e^t$.

Show that:

a) $\displaystyle x\frac{dy}{dx} = \frac{dy}{dt}$ - this was easy enough, could do this.

b) $\displaystyle x^2 \frac{d^2y}{dx^2} = \frac{d^2y}{dt^2} - \frac{dy}{dt}$

Here's how far I got:

Using part a

$\displaystyle \frac{d^2y}{dt^2} = \frac{d}{dt}(x\frac{dy}{dx})$

$\displaystyle \frac{d^2y}{dt^2} = \frac{dy}{dx}\frac{dx}{dt} + x\frac{d}{dt}(\frac{dy}{dx})$

$\displaystyle \frac{d^2y}{dt^2} = \frac{dy}{dt} + x\frac{d}{dt}(\frac{dy}{dx})$

I have no idea what to do with the last, $\displaystyle x\frac{d}{dt}(\frac{dy}{dx})$ part, any help is greatly appreciated

Thanks in advance