A derivative problem Let f (0) = 0, then the necessary and sufficient condition for f (x) to be differentiable at x = 0 is D. Lim H - > 0 (f (2H) - f (H)) / h exists D is wrong. The answer says that D can only guarantee the existence of the left limit, but I think the answer only shows that the left exists, but does not explain why the right does not exist. Who can give a counterexample

A derivative problem Let f (0) = 0, then the necessary and sufficient condition for f (x) to be differentiable at x = 0 is D. Lim H - > 0 (f (2H) - f (H)) / h exists D is wrong. The answer says that D can only guarantee the existence of the left limit, but I think the answer only shows that the left exists, but does not explain why the right does not exist. Who can give a counterexample

If Lim H - > 0 (f (2H) - f (H)) / h exists and f (0) = 0, it can not be obtained that f (x) is differentiable at x = 0, or even continuous
For example: F (x) = 1 (x is not equal to 0)
Lim H - > 0 (f (2H) - f (H)) / h = 0 exists, but f (x) is not differentiable at x = 0. The left and right derivatives do not exist and are discontinuous at x = 0

A derivative problem, not difficult~ Given that the tangents of the curve f (x) = 1 / 6x ^ 2 - 1 and G (x) = 1 + x ^ 3 at x = x0 are perpendicular to each other, find the value of x0?

f'(x)=x/3
g'(x)=3x ²
Vertical slope multiplied = - 1
So x0 / 3 * 3x0 ²=- one
x0 ³=- one
x0=-1

A question about derivative! most urgent! I'd like to ask why this question is related to f (2) ˆ n) Establish contact! Is the first step of the second question!

The first step of the second question is actually two skills, but it's hard to think of
He used a subsequence f (2 ^ n) of F (x) to prove that as long as it is proved that the solution of F (2 ^ n) > = a is not (0, positive infinity), then the solution of F (x) is naturally not (0, positive infinity)
Then he divides the equation represented by F (2 ^ n) into two terms, assuming that if both terms are

Derivative knowledge structure frame diagram! Who can provide the knowledge structure block diagram of high school function derivatives? It's the kind of framed, not the kind of knowledge points ① ② ③ listed like this... It's similar to the flow chart! What about our unit summary~

Go to the library to find the high school mathematics knowledge network (Science), the first page is

Is the derivative of the integrand a definite integral

If f '(x) = f (x)
∫f(x)dx=F(x)+c

What does it mean that the derivative R '(T) of a vector function is equal to zero What is the geometric meaning of this point? That is, the tangent and normal vectors in high numbers are not required to be 0, but what about the geometry when it is zero

If R is displacement, the derivative of vector function R '(T) represents the instantaneous velocity at this time, and equal to 0 means that the instantaneous velocity is 0