The relationship between monotone increasing function and derivative?

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• 1. On monotone function (derivative) Well, If a function is known to be an increasing function, is its derivative greater than or equal to 0 or greater than 0? Given that the function is a decreasing function, then its derivative is less than or equal to 0 or less than 0? Is the derivative of a function greater than 0 an increasing function? Is derivative less than 0 a decreasing function? Is the derivative of a function greater than or equal to 0 an increasing function? Is derivative less than or equal to 0 a decreasing function?
• 2. Let f (x) = e ^ x-1-x-ax ^ 2 (1) if a = 0, find the monotone interval of F (x). (2) if x is greater than or equal to 0, f (x) is greater than or equal to 0 Thank you for your help Thank you for the process
• 3. Find the derivative of this implicit function Let y = sin (x + y), x = π, find y ' The answer in this book is - 1 / 2, how to calculate Let y = sin (x + y) determine the implicit function y = y (x), then dy = (2) How did this "2" come from? Can we do that? Y = sin (π + y) → y = - Sin y → derivation on both sides y '= - y'cosy reduces y' to cosy = - 1, but in this way, y '= cos (x + y) * (1 + y') = cos (π + y) * (1 + y ') = - cosy * (1 + y') is sorted out and y '= - cosy / (1 + cosy) is substituted into cosy = - 1, and the result is y' = 1 / O. what's wrong with this process?
• 4. How to find the derivative of implicit function? For the derivative of the implicit function e ^ y + xy-e = 0, both sides of the equation derive from X: D / DX (e ^ y + xy-e) = e ^ y (dy / DX) + y + X (dy / DX). Why is x (dy / DX) derived from e? Shouldn't it be 0? There is another derivative from y ^ 2-2xy + 9 = 0: 2yy '- 2Y + 2XY' = 0? Why does 9 become 2XY '? Where do you get 2XY' from? I don't understand```````````````
• 5. On the derivative of implicit function X ^ 2 + y ^ 2 = 25 derivative: 2x + 2yy '= 0 question: why is the derivative of x ^ 2 2x and the derivative of Y ^ 2 2yy'? Although the examples in the book are all solved in this way, I still don't understand. Please help me explain it. Thank you
• 6. Finding the derivative of implicit function y to X Y ^ 2-3xy + 9 = 0 requires steps
• 7. Finding the derivative of the implicit function y = y (x) determined by y = x + LNY
• 8. Derivative of implicit function x + y-e ^ x * y = 0
• 9. Finding the derivative of implicit function y = y (x) determined by e ^ y = cos (x + y)
• 10. Finding the second derivative of implicit function (y ^ 2) x = e ^ (Y / x)
• 11. Functions with derivatives less than 0 are monotone increasing A right B wrong
• 12. ）To find the derivative, = (a ^ x) / (a ^ x-1) = 1 + 1 / (a ^ x-1) because a ^ X-1 > 0 is constant, so the derivative is greater than 0, and the function increases monotonously. I don't understand the derivative
• 13. Why the first derivative of a function is always greater than 0 can not deduce the monotonic increasing of the function Our postgraduate entrance examination teacher said that if the first derivative is greater than 0, we can deduce the monotone function, but we can't deduce the increasing or decreasing function, which is different from what I learned before
• 14. It is said in the mathematics book that the derivative is greater than 0 and the function increases monotonically. I think that no matter what the case is, first the derivative is greater than or equal to 0, and then exclude the case that the derivative is in a section or constant to 0 (when the original function is parallel to the X axis, it is not tenable). Therefore, I think what is said in the book is not accurate,
• 15. It is proved that the derivative of monotone function is not necessarily monotone function
• 16. In a hurry Using derivative to solve the problem of monotone interval greater than or equal to 0 Which is greater than or equal to 0 and which is greater than or equal to 0 when the derivative is used to calculate the monotone increasing interval and the value range when an interval is increasing? Why? If the monotone interval is greater than or equal to 0, then the single increasing and single decreasing interval does not have to use the closed interval
• 17. The derivative of a function at a given point: for example, y = x square, x0 = 2
• 18. The derivative of the negative half power of the function y = 3x at point x0 = is equal to
• 19. When Lagrange multiplier method is used to find the extremum of multivariate function, if the solution of partial derivative equal to zero is vector x0, can we use Hessian matrix to determine whether point x0 is the extremum of function?
• 20. Derivative f '(x) is the extreme value of F (x) at x = x0. What does x in x = x0 mean? Does it mean a specific number or a variable?