DO NOW:  Find the equation of the line tangent to the curve f(x) = 3x 2 + 4x at x = -2

3.1 – DERIVATIVE OF A FUNCTION

Remember…  The slope of the line tangent to a curve with equation y = f(x) at x = a:  Similarly, the velocity of an object with position function s = f(t) at t = a:

The Derivative DEFINITION  The derivative of a function f at a number x=a, denoted by f’(a), is If this limit exists.  Also written:

Example 1  Find the derivative of the function f(x) = x 2 – 8x + 9 at the number a.

Interpretation as Slope of Tangent  The tangent line to y = f(x) at (a, f(a)) is the line through (a, f(a)) whose slope is equal to f’(a), the derivative of f at a.  The next two slides illustrate this interpretation of the derivative:

Slope of Tangent (2 interpretations)

Tangent Line Through a Point  Using the point-slope form of the equation of a line, we can write an equation of the line tangent to the curve y = f(x) at point (a, f(a)): y – f(a) = f’(a)(x – a)

Example 2  Find an equation of the line tangent to f(x) = x 2 – 8x + 9 at the point (3,-6)

Interpretation as Rate of Change  The derivative f’(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a.  Note that when the derivative is…  Large, the y-values change rapidly;  Small, the y-values change slowly.

Velocity and Speed  Position Function: s = f(t)  Position along straight line  Velocity Function: f’(a)  Velocity of the function at t = a  Speed: |v(t)|

Example 3  The position of a particle is given by the equation of motion s = f(t) = 1/(1 + t), where t is measured in seconds and s in meters.  Find the velocity and speed after 2 seconds.

Derivative  So far we have considered the derivative of a function f at a fixed number a:  Now we change our point of view and let the number a vary:

Notation y’ “y prime”Nice and brief, but does not name the independent variable “dy dx” or “the derivative of y with respect to x” Names both variables and uses d for derivative. “df dx” of “the derivative of f with respect to x” Emphasizes the function’s name “d dx of f at x” or “the derivative of f at x; Emphasizes the idea that differentiation is an operation performed on f.

Example 4  At right is the graph of a function f. Use this graph to sketch the graph of the derivative f’(x).

Example 4 (solution)

Example 5  For the function f(x) = x 3 – x  Find a formula for f’(x)  Compare the graphs of f and f’

Example 5 (solution)

One-Sided Derivatives  A function y = f(x) is differentiable on a closed interval [a,b] if it has a derivative at every interior point of the interval, and if the limits [the right-hand derivative at a] [the left-hand derivative at a]  Exist at the endpoints.

Example 6  Show that the following function has left-hand and right-hand derivatives at x=0, but no derivative there.

Example 6 (solution)

Practice:  Pg. 205 #1-19odd