How to understand the derivative y to X and X to y

How to understand the derivative y to X and X to y

First definition
Let a function be defined in a neighborhood of a point. When the independent variable has an increment at (also in the neighborhood), the function obtains the increment correspondingly. If the limit exists at the same time, the function is said to be differentiable at the point, and the limit value is called the derivative of the function at the point,
Derivative function
If a function is differentiable at every point in the open interval I, it is said that the function is differentiable in the interval I. at this time, the function corresponds to a definite derivative for every definite value in the interval I, which forms a new function. This function is called the derivative of the original function. It is recorded as. Derivative, which is abbreviated as derivative
Geometric meaning
Geometric meaning of derivative
Geometric meaning of the derivative of a function at a point: the slope of the tangent of the function curve at a point (the geometric meaning of the derivative is the tangent slope of the function curve at this point)
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How to find the derivative of y = | x |?
It should be derivative.

1. When x is greater than 0, y derivative = 1
2. When x is less than 0, y derivative = - 1
3 remove the non differentiable point (0,0), that is, y is not differentiable at (0,0)

Find the derivative of y = (1 - √ x) (1 + 1 / √ x)

y=1-√x+1/√x-1
=-√x+1/√x
=-x^(1/2)+x^(-1/2)
So y '= - 1 / 2 * x ^ (- 1 / 2) - 1 / 2 * x ^ (- 3 / 2)
=-1/(2√x)-1/[2x√x]
=-(x+1)/(2x√x)