Chapter 3 The Derivative Definition, Interpretations, and Rules

Average Rate of Change  For y = f(x), the average rate of change from x = a to x = a+h is

Average Rate of Change, cont.  Graphically, the average rate of change can be interpreted as  the slope of the secant line to the graph through the points (a, f(a)) and (a+h, f(a+h)).

Instantaneous Rate of Change  If y = f(x), the instantaneous rate of change at x = a is

The Derivative  For y = f(x), we define the derivative of f at x, denoted f’(x), to be

Interpretations of the Derivative  The derivative of a function f is a new function f’. The derivative has various applications and interpretations, including:  1. Slope of the Tangent Line to the graph of f at the point (x, f(x)).  2. Slope of the graph of f at the point (x, f(x))  3. Instantaneous Rate of Change of y = f(x) with respect to x.

Differentiation  The process of finding the derivative of a function is called  differentiation.  That is, the derivative of a function is obtained by  differentiating the function.

Nonexistence of the Derivative  The existence of a derivative at x = a depends on the existence of a limit at x = a, that is, on the existence of

Nonexistence, cont.  So, if the limit does not exist at a point x = a, we say that the function f is  nondifferentiable at x = a, or f’(a) does not exist.  Graphically, this means if there is a break in the graph at a point, then the derivative does not exist at that point.

Nonexistence, cont.  There are other ways to recognize the points on the graph of f where f’(a) does not exist. They are  1. The graph of f has a hole at x = a.  2. The graph of f has a sharp corner at x = a.  3. The graph of f has a vertical tangent line at x = a.

Finding or approximating f’(x).  We have seen three different ways to find or apoproximate f’(x). They are;  1. Numerically, by computing the difference quotient for small values of x.  2. Graphically, by estimating the slope of a tangent line at the point (x, f(x)).  3. Algebraically, by using the two-step limiting process to evaluate

Derivative Notation  Given y = f(x), we can represent the derivative of f at x in three ways;  1. f’(x)  2. y’  3.dy/dx

Derivative Rules  Derivative of a Constant Function Rule  If y = f(x) = C, then  f’(x) =0  In words, the rule can be stated;  The derivative of any constant function is 0.

Derivative Rules, cont.  Power Rule

Rules, cont.  Constant Times a Function Rule  If y = f(x) = ku(x), then  f ‘(x) = ku’(x)  In words, the rule can be stated;  The derivative of a constant times a differentiable function is the constant times the derivative of the function.

Rules, cont.  Sum and Difference Rule

Rules, cont.  Product Rule In words, the derivative of the product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

Rules, cont. QQQQuotient Rule In words, the derivative of the quotient of two functions is the bottom function times the derivative of the top function minus the top function times the derivative of the bottom function, all divided by the bottom function squared.

Rules, cont.  General Power Rule

Limits  We write If the functional value f(x) is close to the single real number L whenever x is close to, but not equal to, c.

Limits, cont.  We write And call K the limit from the left if f(x) is close to K whenever x is close to c, but to the left of c.

Limits, cont.  And we write And call L the limit from the right if f(x) is close to L whenever x is close to c, but to the right of c.

On the Existence of a Limit  In order for a limit to exist,  the limit from the left and the limit from the right  must exist and be equal.

Properties of Limits  Let f and g be two functions, and assume that,