What is the specific meaning of "O (△ x)" in the definition of "differential"?

What is the specific meaning of "O (△ x)" in the definition of "differential"?

Δ y = f '(x) △ x + ο (△ x) means that the small increment of function y = f (x) can be expressed as the linear principal part f' (x) △ x plus an infinitesimal of higher order than △ X. in this way, when △ x tends to 0, the infinitesimal of higher order is omitted, and the increment of function can be expressed as dy = f '(x) DX, which is the definition of infinitesimal

What does non differentiable mean

For functions of one variable, differentiability and differentiability are equivalent, and differentiable functions look smooth without sharp points
But for multivariate functions, differentiability is a strong concept, that is to say,
When all the independent variables have an increment, the full increment of the function can be expressed as two parts
One part is that each independent variable is a linear function of increment, the other part is a high-order infinitesimal when the point composed of each increment tends to the origin
The sufficient condition to judge the differentiability of multivariate function is to see if the partial derivative function is continuous

How to logarithm a function?
(cosx) ^ (1 / x) how to logarithm it into e ^ (1 / X * incosx), don't understand?

First of all, we need to understand this: ① the basic algorithm of Ln (x ^ y) = y * LNX logarithm
② X = e ^ (LNX)
And then it's easy
(cosx)^(1/x)=e^( ln (cosx)^(1/x) )
It can be seen from (1) that the exponential part ln (cosx) ^ (1 / x) = 1 / X * incosx
Put it back again
answer all the questions