# Math Help - If x=e^t, what is d^2 y/dx^2 ?

1. ## If x=e^t, what is d^2 y/dx^2 ?

Given $x=e^t$, what is the expression for $\frac{d^2 y}{dx^2}$?

I need it to solve a second order differential equation.

I get $\frac{dy}{dx}=e^{-t} \frac{dy}{dt}$, but when i tried to differentiate the expression $e^{-t} \frac{dy}{dt}$ i am stuck at d/dx (dy/dt), help me out please.

2. Originally Posted by ssadi
Given $x=e^t$, what is the expression for $\frac{d^2 y}{dx^2}$?
I need it to solve a second order differential equation.
I get $\frac{dy}{dx}=e^{-t} \frac{dy}{dt}$, but when i tried to differentiate the expression $e^{-t} \frac{dy}{dt}$ i am stuck at d/dx (dy/dt), help me out please.
$\frac{d^2 y}{dx^2} = \frac{d}{dx} \left[\frac{dy}{dx} \right] = \frac{d}{dt} \left[\frac{dy}{dx} \right] \cdot \frac{dt}{dx}$.

Now note that $\frac{d}{dt} \left[\frac{dy}{dx} \right] = \frac{d}{dt} \left[e^{-t} \cdot \frac{dy}{dt}\right] = - e^{-t} \frac{dy}{dt} + \frac{d^2 y}{dt^2} e^{-t}$.

3. Originally Posted by mr fantastic
$\frac{d^2 y}{dx^2} = \frac{d}{dx} \left[\frac{dy}{dx} \right] = \frac{d}{dt} \left[\frac{dy}{dx} \right] \cdot \frac{dt}{dx}$.

Now note that $\frac{d}{dt} \left[\frac{dy}{dx} \right] = \frac{d}{dt} \left[e^{-t} \cdot \frac{dy}{dt}\right] = - e^{-t} \frac{dy}{dt} + \frac{d^2 y}{dt^2} e^{-t}$.
Ok, got the required form. Thanks for the helping hand