What is the difference between dy/dx and f `(x)?

Question

In general, what is the difference between dy/dx and f ` (x)?

How about in this situation: "Find dy/dx" y= (x^3)(sin(5x)) ?

I presume there is no difference.

In what situation do they differ?

Answer 1

There is no difference in your case, because it's just notation. but in general (dy/dx) is probably the best notation. In both cases f'(x) and (dy/dx) are the same, because it just means find the derivative in respect to x. When you get to higher level math classes, you'll see why (dy/dx) is the best notation to use.

Answer 2

In calculus, they mean the same thing. However, dy and dx are actual values, meaning infinitesimal changes in y and x, respectively. those values can be used in other calculations, whereas f'(x) cannot. An example is dy/dx = 1, so dy = dx. If you have f'(x) = 1, it implies dy/dx = 1, but you cannot go any further with f'(x) in calculations.

Answer 3

Let's take an example of y = ax² + bx + c. If you call
f(x) = ax² + bx + c then you can say y = f(x). In the first case it makes sense to talk about dy/dx = 2ax + b and in the 2'nd case it's a bit more compact to say f'(x). But they both refer to the same thing; Namely the 'slope' of a tangent line to the parabola
y = ax² + bx + c. You really should be 'comfortable' with both forms as they do kinda get used interchangeably. I've always liked the
dy/dx notation because it 'looks' like rise (change in y) over run (change in x).
But then, I learned this stuff 'bout 50 years ago, so maybe I'm just gettin' old and set in my ways ☺

Doug

Answer 4

As long as your not doing implicit differentiation
dy/dx and f'(x) are interchangeable. It means y is a function
of f. Say you had (y^2 +x^2)*xy. You can still differentiate
and find slopes and tangent lines, it's just not starting from
a y = f(x) situation.

Answer 5

The dX/dY notation is from Leibniz and f'(x) is from Newton.

In general, the Newton notation is more convenient if it can be used. For more complex problems, it may be difficult to express the problem with Newton's notation.

Leibniz's notation is messier but more powerful.

Answer 6

It's the same unless you have many variables

Example: f(x,y,z) = x+2y+z^2

In this case, if you say df/dy it's clear that x and z are "constant" and you are actually finding one of the partial derivative of f.

df(x,y,z)/dy = 0 + 2 + 0 = 2

df(x,y,z)/dz = 1

df(x,y,z)/dz = 2z

Ilusion

Answer 7

when having f'(x) means you have to integrate the functions for x
you have to be carefull with dx/dy and dy/dx..... depending on which one you have it would determine whether you have to integrate for x or y

Answer 8

You will understand once you get to rate changes and word problems. Don't worry right now.

You can assume it's f'(x)