The derivate of zero - as well as the derivative of ANY constant (non-variable) number, is zero. (A graph of y = 0 for example will be a horizontal line - the slope is zero.)
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If you mean "what is the deriviative of f(x) = 0?", the answer is 0. (zero) The deriviative of any constant is zero.
The signum function, also known as the sign function, is not differentiable at zero. This is because the derivative of the signum function is not defined at zero due to a sharp corner or discontinuity at that point. In mathematical terms, the signum function has a derivative of zero for all values except at zero, where it is undefined. Therefore, the signum function is not differentiable at zero.
Yes.
The rate of change of any function is its derivative. The equation of a horizontal line is simply a constant, for example y=10. The derivative of any constant is ZERO.
You take the derivative of the function. The derivative is another function that tells you the slope of the original function at any point. (If you don't know about derivatives already, you can learn the details on how to calculate in a calculus textbook. Or read the Wikipedia article for a brief introduction.) Once you have the derivative, you solve it for zero (derivative = 0). Any local maximum or minimum either has a derivative of zero, has no defined derivative, or is a border point (on the border of the interval you are considering). Now, as to the intervals where the function increase or decreases: Between any such maximum or minimum points, you take any random point and check whether the derivative is positive or negative. If it is positive, the function is increasing.