What are the zeros of a derivative function?
To find zeros of the derivative, look at the graph of the derivative function. The zeros will be the points at which the derivative crosses the x-axis. Using a graphing calculator’s trace button, you can find the exact locations of x when the function is 0.
Is the domain of the derivative always the same as the function?
No, because when you are given a differentiable function, you must be given a domain over which the function is defined. Then the domain of the derivative can be no greater than the domain of the function. Consider the graph below which graphs the function over the domain .
Which function has its derivative equal to itself?
the Exponential Function
A derivative of the Exponential Function: We have only one function which is an exponential function whose derivative equals itself.
How do you know if a function has a derivative?
The derivative of a function at a given point is the slope of the tangent line at that point. So, if you can’t draw a tangent line, there’s no derivative — that happens in cases 1 and 2 below.
What does it mean when the derivative is undefined?
If there derivative can’t be found, or if it’s undefined, then the function isn’t differentiable there. So, for example, if the function has an infinitely steep slope at a particular point, and therefore a vertical tangent line there, then the derivative at that point is undefined.
Why is the derivative of a function always 0?
The derivative represents the change of a function at any given time. The constant never changes—it is constant. Thus, the derivative will always be 0. Consider the function x2 −3. It is the same as the function x2 except that it’s been shifted down 3 units. The functions increase at exactly the same rate, just in a slightly different location.
Why is the derivative of x2 the same as x2?
The constant never changes—it is constant. Thus, the derivative will always be 0. Consider the function x2 −3. It is the same as the function x2 except that it’s been shifted down 3 units. The functions increase at exactly the same rate, just in a slightly different location. Thus, their derivatives are the same—both 2x.
How to find the derivative of a constant with x0?
Since any constant can be written in terms of x0, finding its derivative will always involve multiplication by 0, resulting in a derivative of 0. Use the limit definition of the derivative: