Recall that the derivative of a constant times a function is the constant times the derivative of the function, and the derivative of a sum of two functions is the sum of the two derivatives. Because of this the derivative of the difference of two functions is the difference of the two derivatives. Does this continue to work for products? Specifically, is the derivative of the product of two functions the product of the two derivatives?[br][br]In the app, enter desired formulas for [i]f[/i]([i]x[/i]) and [i]g[/i]([i]x[/i]). The product of the two functions is [i]s[/i]([i]x[/i]). All three functions are graphed, [i]f[/i] in green, [i]g[/i] in blue, and [i]p[/i] is red.[br][br]Check the checkbox for Tangent Lines to see the tangent lines and their slopes (the derivatives at [i]x[/i] = [i]a[/i]) illustrated. Check the checkbox for Values to see the values of the three functions and their derivatives at [i]x[/i] = [i]a[/i]. Compare the values to see if the conjectured relationship holds.

Alas, the derivative of a product is not the product of the derivatives. Check the checkbox for Product Rule to see the actual product rule. The Product Rule is then applied at the formula level and at the numerical level for [i]x[/i] = [i]a[/i].