Derivative of the Cubic Function f(x)=x^3

This applet demonstrates the step-by-step process of finding the derivative of the cubic function f(x)=x^3. Students can interactively explore how the derivative is derived both algebraically using differentiation rules and geometrically through dynamic visual representation. The applet simultaneously displays the graph of the original cubic function alongside the graph of its derivative f'(x)=3x^2, allowing learners to observe the direct correspondence between the two curves. A key feature of this applet is its visualization of the geometric meaning of the derivative. As students manipulate points along the cubic curve, they can watch the tangent line change in real time. The slope of each tangent line is displayed numerically, and the locus of these slope values traces out the parabola representing the derivative function. This dynamic interaction powerfully illustrates that the derivative at any point on the curve equals the instantaneous rate of change or the slope of the tangent line at that point. The applet combines algebraic computation with geometric intuition, reinforcing the fundamental calculus concept that differentiation is a process of finding rates of change. Students can verify the power rule for differentiation, observing that applying the rule d/dx[x^n]=nx^(n-1) to x^3 yields 3x^2, which matches the graph produced dynamically. This dual representation—symbolic and visual—supports multiple learning pathways and helps bridge the gap between abstract formulas and concrete graphical understanding. This interactive resource is particularly valuable for students studying introductory calculus, as it provides immediate visual feedback that reinforces correct reasoning and helps identify misconceptions about the relationship between a function and its derivative. It serves as an excellent supplementary tool for classroom instruction or independent study of differentiation techniques for basic elementary functions.