Dynamic Slope of Derivative

The following graphic traces the equation of the derivative of the f(x) = x2. Drag the slider a to move the cursor around the curve and S will draw the equation of the derivative. You can check that the derivative is correct by entering g(x) = 2x into the input bar. If the line that is generated is the same as the trace generated by S, then our calculation is correct!
You can change the function being displayed in the above graphic, by typing f(x) = new function into the input bar, where new function is whatever function you wish to graph. Use Ctrl+f to clear the current trace. Try graphing the following functions and then trace the equation of their derivatives (a) f(x) = x^3 (which represents x3, as the ^ notation means 'to the power of' (b) f(x) = 1 - x^2 (c) f(x) = 2x^2 + 3 (d) f(x) = x^3 - 6x Try to come up with a conjecture about the relationship between the graphs of the functions and those of their derivatives.