If dy/dx=f ' (x), what do "dy", "dx", "d/dx", and "d/dy" mean individually? Please help.?

Question

I'm working on integration and my teacher's notes involved the breakdown of dy/dx with terms like dx, dy, d/dy, d/dy, dx/d, dy/d (not all of those, but I'm assuming the combinations are possible...maybe) and I don't understand exactly what they mean, verbally or conceptually.

All I know (assuredly) is dy/dx is equal to the derivative of the function f(x) in terms of x and y. PLEASE HELP!

Answer 1

When we say dy/dx or f '(x), we mean a ratio

dy/dx = f '(x) = [f(x+h)-f(x)]/h as h --> 0

This is the differential co-efficient of the function f(x) w.r.t.x

d/dx means the same but d/dy means the differential of a function w.r.t.y

dx and dy means differentials only. You can refer to topic of differentials in the calculus for study of differentials.

There is nothing in calculus like dy/d and dx/d

Answer 2

For the purpose of calculus, if you have a function f(x) = y,



The derivative of the function or gradient function can be shown by



dy/dx (Leibniz notation) and f'(x) (prime notation).



The 'd' infront of x or y stands for 'infintissimal change in', which relates to rate of change. If the difference in x2 and x1 becomes really small the rate of change approaches the rate of change function, as shown in the limit h->0 theorem (where h is the change in x) to find derivative from first principle.





So in integration dy means really small change in y. This can be used to do integration in parts.

Also, you may see that to find the derivative of say x^2 on its own, the notation is d/dx(x^2). This does not have much meaning in terms of actual values placed on the letters, it is more of an operator just used to find 'derivative with respect to x'.



Hope this helps.

Answer 3

d/dx is the derivative with respect to x. So that means you treat your other variables as constants and take the derivative.

d/dy, not surprisingly then is the derivative with respect to y.

dy/dx is taking the derivative of function y with respect to x. So if you have a function y = 3x + 1

dy/dx = d/dx ( 3x + 1)

= 3



So let's say you have a function f(x,y,z) = x^2 + y^2 + z^2

d/dx(f(x,y,z)) = df(x,y,z) / dx = d/dx (x^2 + y^2 + z^2)

= 2x since y^2 and z^2 are considered constants

similarly,

df(x,y,z) / dz = d/dz (x^2 + y^2 + z^2)

= 2z



dy/d and dx/d are not actual notations.

Answer 4

the combinations you will see are d/dx, d/dy, and dx/dy. These don't mean anything individually, they are just notation. d/dx is the derivative in terms of x, d/dy is the derivative in terms of y, and dx/dy is the derivative in terms of both x and y.

Answer 5

"dy" = "differential" y, an infinitesimal, or vanishingly small, incremental change in y

"dx" = "differential" x, an infinitesimal, or vanishingly small, incremental change in x

"d/dx" = a derivative operator. Its operation is the form the ratio of the differential of the target of the operation with the differential of x

"d/dy" = a derivative operator. Its operation is to form the ratio of the differential of the target of the operation with the differential of y.



dy/dx, commonly called the derivative of y, is actually the ratio of the differential of y, "dy", to the functionally related differential of x, "dx".