Section 2.2 – Basic Differentiation Rules and Rates of Change

The Constant Rule A constant function has derivative, or: Note: The constant function is a horizontal line with a constant slope of 0.

Examples Differentiate both of the following functions. The function is a horizontal line at y = 13. Thus the slope is always 0. The function is a horizontal line because e, Pi, and i all represent numeric values. Thus the slope is always 0.

The Power Rule For any real number n, the power function has the derivative, or: Ex:

Examples Differentiate all of the following functions. 1. Bring down the exponent2. Leave the base alone3. Subtract one from the original exponent. The procedure does not change with variables. Make sure the function is written as a power function to use the rule.

The Constant Multiple Rule The derivative of a constant times a function, is the constant times the derivative of the function. In other words, if c is a constant and f is a differentiable function, then Objective: Isolate a power function in order to take the derivative. For now, the cf(x) will look like.

Examples Differentiate all of the following functions. “Pull out” the coefficient Make sure the function is written as a power function to use the rule. Take the derivative Make sure the function is written as a power function to use the rule.

The Sum/Difference Rule The derivative of a sum or a difference of functions is the sum or difference of the derivatives. In other words, if f and g are both differentiable, then OR Objective: Isolate an expressions in order to take the derivative with the Power and Constant Multiple Rules.

Example 1 If find if and. Find the derivative of k Evaluate the derivative of k at x = 5

Example 2 Evaluate: Sum and Difference Rules Constant Multiple Rule Power Rule Simplify

Example 3 Find if. First rewrite the absolute value function as a piecewise function Since the one-sided limits are not equal, the derivative does not exist at the vertex Find the Left Hand Derivative Find the Right Hand Derivative

Example 4 Find the constants a, b, c, and d such that the graph of contains the point (3,10) and has a horizontal tangent line at (0,1). What do we know: 1. f(x) contains the points (3,10) and (0,1) 2. The derivative of f(x) at x=0 is 0 Use the points to help find a,b,c We need another equation to find a and b Find the Derivative We know the derivative of f(x) at x=0 and x=1 is 0

Example 4 (Continued) Find the constants a, b, c, and d such that the graph of contains the point (3,10) and has a horizontal tangent line at (0,1). What do we know: 1. f(x) contains the points (3,10) and (0,1) 2. The derivative of f(x) at x=0 is 0 AND Find aUse the Derivative a = 1, b = 0, and c = 1

Derivative of Cosine Using the definition of derivative, differentiate f(x) = cos(x). Find the derivative of sine if you have time

Derivative of Sine Using the definition of derivative, differentiate f(x) = sin(x).

Derivatives of Sine and Cosine

Example 1 Differentiate the function: Sum and Difference Rules Constant Multiple Rule Power Rule AND Derivative of Cosine/Sine Simplify Rewrite the ½ to pull it out easier Rewrite the radical to use the power rule

Example 2 Find the point(s) on the curve where the tangent line is horizontal. Horizontal Lines have a slope of Zero. First find the derivative. Find the x values where the derivative (slope) is zero Find the corresponding y values

Calculus Synonyms The following expressions are all the same: Instantaneous Rate of Change Slope of a Tangent Line Derivative DO NOT CONFUSE AVERAGE RATE OF CHANGE WITH INSTANTANEOUS RATE OF CHANGE.

Position Function The function s that gives the position (relative to the origin) of an object as a function of time t. Our functions will describe the motion of an object moving in a horizontal or vertical line s(t) t Origin TIME: Description of Movement: UpwardDownwardNo Movement

Displacement Displacement is how far and in what direction something is from where it started after it has traveled. To calculate it in one dimension, simply subtract the final position from the initial position. In symbols, if s is a position function with respect to time t, the displacement on the time interval [ a, b ] is: s(t) t EX: Find the displacement between time 1 and 6.

Average Velocity The position function s can be used to find average velocity (speed) between two positions. Average velocity is the displacement divided by the total time. To calculate it between time a and time b : s(t) t EX: Find the average velocity between time 2 and 5. Two Points Needed It is the average rate of change or slope.

Instantaneous Velocity The position function s can be used to find instantaneous velocity (often just referred to as velocity) at a position if it exists. Velocity is the instantaneous rate of change or the derivative of s at time t : s(t) t EX: Graph the object’s velocity where it exists v(t) t Slope = 2 Corner Slope = 0 Slope = -4 Slope = -2 Slope = 0 Slope = 4 One Point Needed

Position, Velocity, … Position, Velocity, and Acceleration are related in the following manner: Position: Velocity: Units = Measure of length (ft, m, km, etc) The object is… Moving right/up when v(t) > 0 Moving left/down when v(t) < 0 Still or changing directions when v(t) = 0 Units = Distance/Time (mph, m/s, ft/hr, etc) Speed = absolute value of v(t)

Example 1 The position of a particle moving left and right with respect to an origin is graphed below. Complete the following: 1.Find the average velocity between time 1 and time 4. 2.Graph the particle’s velocity where it exists. 3.Describe the particles motion s(t) t

Example 2 Sketch a graph of the function that describes the motion of a particle moving up and down with the following characteristics: The particle’s position is defined on [0,10] The particle’s velocity is only positive on (4,7) The average velocity between 0 and 10 is 0.

Example 3 The position of a particle is given by the equation where t is measured in seconds and s in meters. (a) Find the velocity at time t. The derivative of the position function is the velocity function.

Example 3 (continued) The position of a particle is given by the equation where t is measured in seconds and s in meters. (b) What is the velocity after 2 seconds? (c) What is the speed after 2 seconds? m/s

Example 3 (continued) The position of a particle is given by the equation where t is measured in seconds and s in meters. (d) When is the particle at rest? The particle is at rest when the velocity is 0. After 1 second and 3 seconds