Writing derivatives

Question

Can somebody please check my working? The variable $y$ depends on $x$ and $x$ and $t$ are related by $x=e^t$

Show that $x\frac{dy}{dx}=\frac{dy}{dt}$

$$x=e^t$$ $$dx=e^t dt$$ $$\frac{dx}{dt}=e^t=e^t\times1=e^t \times\frac{dy}{dy}$$ $$\frac{dx}{dt}\times\frac{dy}{dx}=x\frac{dy}{dy}\times\frac{dy}{dx}$$ $$\frac{dy}{dt}=x\frac{dy}{dx}$$

Is this correct?

Answer 1

In your answer, I am not sure how to get from line (3) to line (4).

I would proceed using the Chain Rule: $$ \frac{dy}{dt} = \frac{dy}{dx} \frac{dx}{dt} $$ but you know how $x$ depends on $t$, can you find $dx/dt$ and plug in?

Answer 2

You are given $$y=f(x),x=e^t.$$

Then

$$\frac{dy}{dx}=f'(x)$$ and by the chain rule,

$$\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}=f'(x)e^t=f'(x)x,$$ hence the claim.