Question

Notations for differentiation

Answer 1

I shouldn’t often use y’ to represent the derivative of a function because y’ does not provide which variable am I performing the differentiation process with respect to. How can y' be used for implicit functions?

Answer 2

Why can't dy/dx be thought of as the quotient of the two infinitesimal quantities dy and dx?

Answer 3

Given two points K and L on a curve, K is the fixed and L varies along the curve. If L gets so close to L but remains distinct from it by an amount dx as the difference in the independent variable x and dy as the change in the dependent variable y. So the slope becomes dy/dx.

Answer 4

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I shouldn’t often use y’ to represent the derivative of a function because y’ does not provide which variable am I performing the differentiation process with respect to. How can y' be used for implicit functions?
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yes, the prime notation requires that we "know" what variables are what.
However, if it is clear (by statement, for example)
then the prime notation is easier to type.

if you had (for example)
y = x*y^2
and we take the derivative with respect to x:
y' = y^2 + 2 x y y'
using the product rule. If it's not clear, you can always switch to d/dx notation:
d/dx y = d/dx ( x*y^2) = y^2*d/dx x + x*d/dx (y^2)