The graph of a function as well as the same function with transformations applied is shown on the left. Compare the derivative of the original function with the derivative of the transformed function. [br][list][*]Use the slider tool for k to adjust the coefficient, which corresponds to a vertical stretch/shrink. [/*][*]Use the slider tool for b to adjust the constant term, which corresponds to a vertical shift. [/*][*]Checkboxes for f(x) and g(x), as well as f'(x) and g'(x), show/hide their graphs. [/*][*]The "tangents" checkbox will show/hide a tangent line segment on each graph. [/*][*]With "tangents" shown, click the "Trace Derivatives" button to trace the graph of the derivative functions on the right. [/*][/list]

As we begin to look at different techniques for differentiation (i.e., finding derivatives), it is important to keep in mind [i]two different types of differentiation rules[/i]. [br][br][b]Derivative Formula:[/b] A rule that simply states the formula for the derivative of a [i]particular function[/i]. For example, [math]\frac{d}{dx}\left[\sin x\right]=\cos x[/math] is a derivative formula. Think of the differentiation operator [math]\frac{d}{dx}[/math] as an operation that takes a function ([math]\sin x[/math]) as an input and produces its derivative function ([math]\cos x[/math]) as an output. [br][br][b]Differentiation Rule: [/b]A rule that states how the differentiation operator "interacts" with function operations (addition, subtraction, multiplication, division, composition). For example, "the derivative of a sum of two functions is the sum of their derivatives," i.e., [math]\frac{d}{dx}\left[f(x)+g(x)\right]=f'(x)+g'(x)[/math]. This rule is true no matter whether f and g are power functions, trig functions, logarithms, etc., as long as they are differentiable. So, the rule is about the differentiation operator and the function operation of addition, not about the specific functions involved.