Question

I'm not sure whether or not it's always okay to manipulate dy and dx algebraically...and derive wrt y (dx/dy) then rearrange and solve for dy/dx

Answer 1

\[x = y^2 \]
\[dx/dy = 2y\]\[dy/dx = 1/2y\]

like this ;o

Answer 2

gives an implicit ans but then plugging y back into it would solve the prob., right

Answer 3

You are allowed to do this, just be aware of 'proper' notation, so to say

\[\frac{\delta x(y)}{\delta y}=2y\]

OR

\[\frac{\delta y(x)}{\delta x}=\frac{1}{2y}\]

Answer 4

thank you =)

Answer 5

\(\color{blue}{\text{Originally Posted by}}\) @KinzaN
\[x = y^2 \]
\(\color{blue}{\text{End of Quote}}\)
Differentiate both sides with respect to \(x\) and you get: \[
1 =2y\frac{dy}{dx} \implies \frac{dy}{dx} = \frac{1}{2y} \implies dy = \frac{dx}{2y}
\]Differentiate both sides with respect to \(y\) and you get: \[
\frac{dx}{dy} = 2y \implies dx = 2y \;dy
\]You cannot do partial differentiation because there is no function.

Differentials come from using \(dy = \frac{dy}{dx}dx\), which is true due to the chain rule. They are not a result of multiplying both sides of the equation by a differential.

Answer 6

@KinzaN if you're "just" doing physics, you can do whatever the hell you like!! it's only the maths bods that get hung up on this stuff. (jk)

my personal fave is d2x/dt^2 = dv/dt = dv/dt*dx/dx = dx/dt*dv/dx = v dv/dx. now that's magic! and it gives you Newton's third law of motion from thin air....

people play fast and loose with this stuff all the time, eg in the notation for integration parts where dv's and du's are passed round like snuff at a wake. the wheels only really come off with partials.

some very clever people thought calculus was black magic way back then, eg google Bishop George Berkeley vs Newton.