[list][*]Use the input box for f(x) to define a function. Use the slider tools for k and b to adjust the graph of g(x) = k f(x) + b. Observe how the numbers k and b affect the graph of g(x). [/*][*]Use the "tangents" checkbox to show / hide tangent line segments on the graphs of f and g. [/*][*]Click the "Trace Derivatives" button to sketch a graph of the derivative functions f'(x) and g'(x). Use the checkboxes for f'(x) and g'(x) to show / hide their graphs. [/*][/list]

There are basically two types of shortcuts for finding derivatives: derivative formulas and differentiation rules. [br][br]A [b]derivative formula[/b] uses the formula for a simple function (i.e., no extra operations) to tell you the formula of the derivative function. These are rules like [math]\frac{d}{dx}\left[x^3\right]=3x^2[/math] or [math]\frac{d}{dx}\left[\ln x\right]=\frac{1}{x}[/math]. [br][br]A [b]differentiation rule[/b] tells us how the differentiation operator interacts with function operations like addition, subtraction, and constant multiples. (Later we will also learn rules for composition, multiplication, and division.) For example,[br][br][math]\frac{d}{dx}\left[f(x)+g(x)\right]=\frac{d}{dx}\left[f(x)\right]+\frac{d}{dx}\left[g(x)\right]=f'(x)+g'(x)[/math][br][br]Notice that we don't need to know what f and g are to use this rule; it applies to the operation "+", not the particular functions being added together.