The Derivative Definition, Interpretations, and Rules

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))  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