Question

If y = x then dy, the differential of y, as a changes from 64 to 64.1 is given
by

Answer 1

`dy \ = \ 0.1`

Explanation:

Considering the Leibniz notation to represent the derivative of
`y` with respect to
`x`, suppose
`y \ = \ f\left(x\right)` is a differentiable function, let
`dx` be the independent variable such that it can be designated with any nonzero real number, and define the dependent variable
`dy` as

`dy \ = \ f'\left(x\right) \ dx`,

where
`dy` is the function of both
`x` and
`dx`. Hence, the terms
`dy` and
`dx` are known as differentials

Dividing both sides of the equation by
`dy`, yield the familiar expression

`\displaystyle(dy)/(dx) \ = \ f'\left(x\right)`.

Given that
`f\left(x\right) \ = \ x` and
`dx \ = \ 64.1 \ - \ 64 \ = \ 0.1`, hence

`f'\left(x\right) \ = \ 1`.

Subsequently,

`dy \ = \ f'\left(64\right) \ * \ 0.1 \\ \\ dy \ = \ 1 \ * \ 0.1 \\ \\ dy \ = \ 0.1`.