Subsitution $f(x)=y$

Question

I have to solve a third order differential equation.

Not the topic of question, but I was thinking of letting $f(x)$ (which I'm solving for) be equal to $y$, so that I can use $dy$ and $dx$.

If I didn't involve y, would I be using $df$ and $dx$, or $df(x)$ and $dx$, or what, would be my differentials?

Answer 1

You wouldn't be confused so much with representation if you think in terms of what you are actually doing while we find differentials.

For any equation (basically a curve in your graph), you are finding the slope(rise/fall) to figure out where it attains min and max and the curves behavior.

When you do, $\frac{d(x^2)}{dx}$ or $\frac{dy}{dx}$ where $f(x)=x^2,$ or $y=x^2.$ It's all the same. You are just figuring out how the slope of the curve changes with respect to changing values in $x$ axis.

Answer 2

It must be $$\frac{d(f(x))}{dx}$$ Since, $$\frac{dy}{dx}=\frac{d(f(x))}{dx}$$