Find how much limx → ∝ {e ^ (1 / x) + 1 / X} ^ x equals, One step is substitution, let 1 / x = y, y = 0 limy → 0 (e ^ y + Y-1) / y = 2. Why is this step equal to 2

Find how much limx → ∝ {e ^ (1 / x) + 1 / X} ^ x equals, One step is substitution, let 1 / x = y, y = 0 limy → 0 (e ^ y + Y-1) / y = 2. Why is this step equal to 2

Law of lobida
lim (e^y+y-1)/y=lim (e^y+y-1)‘/y’=lime^y+1=2

Limx tends to 0 (a ^ x-1) / x equals

Prove by definition: limx - > infinite sin1 / x = 0

X tends to infinity, 1 / X tends to 0, sin0 = 0

Let y = 2x ^ 3-x-1 (1) when x = 1, △ x = 0.1, find △ Y / △ X; (2) use the domain of definition to find the derivative of the function at x = 1

△y=2(1+0.1)^3-(1+0.1)-1-[2-1-1]=2*1.1^3-0.1
Use the calculator
The second question is to use the definition of derivation
lim △x->0 △y/△x=lim △x->0 [[2(x+△x)^3-x-1]-2x^3-x-1]/△x=6x^2-1
Expand (x + △ x) ^ 3

What is the point where the derivative does not exist and how is it reflected in the derivative function?

The point where the left and right derivatives are not equal or one side derivative does not exist is called the point where the derivative does not exist, which is reflected in the derivative function. That point is meaningless or can not be calculated in the derivative function

Derivative second order differentiable function
What is second order differentiable function
I read some reference materials of derivative, and I don't understand it

First of all, you need to understand what is the first derivative. If the left derivative of a function at a certain point = the right derivative, then the function is differentiable at that point. If the function is differentiable at every point in the domain of definition, then the function is first-order differentiable. The function has first-order derivative and second-order derivative. Similarly, first of all, the second-order derivative f (x) must be first-order differentiable

Derivative of complex function
If we only know the analytic formula of a function in real variable, we can get the derivative without the definition of derivative. So, if we only know the analytic formula of a function in complex variable, we can get the derivative without the definition of derivative?
It's not a composite function

Divide the function into real part and complex part and take the derivative separately
EG：y=2x+i(3x)
y'=2+i(3)

A proof of derivative definition of complex function
Title: using the definition of derivative to discuss whether the derivative of function f (z) = re (z) exists
Note that it is defined by derivative. Don't prove it by Cauchy Riemann equation!

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from morning till night
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Whether the complex variable function and its derivative can be expressed in the same coordinate system
If f (z) = the square of Z, then the derivative of F (z) is equal to 2Z

Let Z be x and f (z) be y

The derivative of e ^ (Xi / 2) to X

How about plural derivation?
Method 1
e^(xi/2)=cos(x/2)+isin(x/2)
[e^(xi/2)]'=-(1/2)sin(x/2)+(i/2)cos(x/2)
Method 2
[e^(xi/2)]'=(i/2)e^(xi/2)
=(i/2)[cos(x/2)+isin(x/2)]
=-(1/2)sin(x/2)+(i/2)cos(x/2)
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