Given a function f that is differentiable on its domain, write and explain the relationship between the differentials dx and dy.

Verified step by step guidance

1. Understand that the differential 'dx' represents an infinitesimally small change in the variable 'x'. It is often used to denote a small increment in the input of a function.
2. Recognize that 'dy' represents the corresponding infinitesimal change in the function 'f(x)' when 'x' changes by 'dx'. This relationship is expressed through the derivative of the function.
3. Recall the definition of the derivative: if 'f' is differentiable at a point 'x', then the derivative 'f'(x) = rac{dy}{dx}', which means 'dy' can be expressed as 'dy = f'(x) imes dx'.
4. Interpret this relationship: the differential 'dy' is directly proportional to 'dx' and scaled by the derivative 'f'(x)', indicating how steep the function is at that point.
5. Use this relationship in applications such as linear approximations, where 'dy' can be used to estimate changes in 'f(x)' based on small changes in 'x'.