Question

Group Activity In Exercises \(27-30\) , use NDER to graph the derivative of the function. If possible, identify the derivative function by looking at the graph. $$y=0.25 x^{4}$$

Short answer

The derivative of the function \(y=0.25x^{4}\) is \(y=x^{3}\)

Step-by-step solution

Compute the derivative

The power rule states that the derivative of a function of the form \(cx^n\) is \(ncx^{n-1}\). Applying this rule to the given function \(y=0.25x^{4}\), the derivative is found to be \(4*0.25x^{4-1} = x^3\).

Graph the derivative function

To graph the function \(y=x^3\), start at the origin (0,0) and plot points for a few positive and negative x-values. The graph would rise to the right, illustrating that the function increases as x increases, and falls to the left, illustrating that the function decreases as x decreases.

Identify the function of the derivative

The graph of \(y=x^3\) is a standard graph of a cubic function. By analyzing the graph, it can be determined that the derivative function is a cubic function of the form \(y=x^3\).