Chapter Two: Section Two Basic Differentiation Rules and Rates of Change

Chapter Two: Section Two Remember from our last section that the complex definition of the derivative will give us:  Either a simple numerical value if we decide in advance a target domain value at which to construct our tangent line or  An algebraic formula that will then find the slope of the tangent line at any domain value for the function.

Chapter Two: Section Two What we need to remind ourselves of constantly, and it is one of the reasons why I am emphasizing the definition of the derivative, is we are finding a slope whenever we talk about the derivative of a function. This idea should inform some of the following differentiation rules that are presented clearly in section two of this chapter.

Chapter Two: Section Two The constant rule – The derivative of a constant is zero.  Why would this be true?  What do we know about functions that are constants, such as the function y = 5?  Why woould the derivative of such a function be zero?

Chapter Two: Section Two The power rule – If n is a rational number, then the function f (x) = x n is a differentiable function and the derivative of the function is nx n-1.  Can you convince yourself that this rule makes sense?  What should the derivative of a linear function look like? What degree would that function be?  Think about the derivative of a parabola. What should it look like according to the power rule? Can you convince yourself that this is true?

Chapter Two: Section Two The constant multiple rule - If f is a differentiable function and c is a constant, then the function cf is a differentiable function whose derivative is simply c times the derivative of f.  Would a similar rule apply to the function f + c ?  Why should this rule work? Think again about the simple parabola whose equation is y = x 2. Why would its derivative be different than the derivative of the function y = 3 x 2 ?

Chapter Two: Section Two The sum and difference rules – The sum or difference of two differentiable functions f and g are themselves differentiable functions. The derivative of the function f + g is simply the sum of the derivatives of f and g. Similarly, the derivative of the difference function f – g is the difference of the derivatives of the individual functions f and g.

Chapter Two: Section Two Finally, you are presented with the rather amazing fact that the cosine and sine curves are related through differentiation as well. We will look at the verification of these facts using our trigonometric identities for angle addition and we will look at this using an EXCEL spreadsheet as well.