Basic Differentiation Rules & Rates of Change Chapter 3.2
Basic Differentiation Rules • In chapter 2, you first learned how to find limits using the definition • Later, we used the definition to prove some general rules (or properties or, more appropriately, theorems) that made finding limits much easier • The same can be done for differentiation • Using the definition of the derivative, we can find a few basic theorems that make differentiation much easier • You MUST know these rules, and the ones in the next section that follow, by heart!!!
Theorem 3.2: The Constant Rule THEOREM: The derivative of a constant function, , where c is a real number, is PROOF Let . By the limit definition of the derivative
Example 1: Using the Constant Rule • , • , • , • ,
Theorem 3.3: The Power Rule THEOREM: If n is a rational number, then the function is differentiable and For f to be differentiable at , n must be a number such that is defined on an interval containing 0. For example, if , then is not defined at , so it is not differentiable at , though it is differentiable everywhere else. Think of this example in terms of theorem 3.1: differentiability implies continuity if and only if discontinuity implies nondifferentiability. Since is not defined at , it is discontinuous at so it is not differentiable at .
Theorem 3.3: The Power Rule THEOREM: If n is a rational number, then the function is differentiable and PROOF By the limit definition of the derivative Simplifying this expression requires that we know the binomial theorem, which I will use here without proof
Theorem 3.3: The Power Rule THEOREM: If n is a rational number, then the function is differentiable and PROOF
Example 2: Using the Power Rule • , • , • , Note that it is generally easier to rewrite radicals in their rational exponent form, and to rewrite variables in the denominator with a negative exponent. With enough practice, you will be able to do these in your head.
Example 3: Finding the Slope of a Graph Find the slope of the graph of when Using the power rule we have . The slope of the tangent line at a) is . The slope at b) is . The slope at c) is .
Example 4: Finding an Equation of a Tangent Line Find an equation of the tangent line to the graph of when . The derivative at a point is the slope of the tangent line to the curve at that point. So finding the equation of the tangent line means using , where is the point on the curve. In this case, so . So the point is . All that we need is the slope and this is provided by the derivative So the slope of the tangent line is . Therefore, the equation of the tangent line is NOTE: on the AP exam, it is never necessary to write a linear equation in form.
Theorem 3.4: The Constant Multiple Rule THEOREM: If f is a differentiable function and c is a real number, then cf is also differentiable and PROOF Using the derivative definition we have
Example 5: Using the Constant Multiple Rule • , • , • , • , • ,
Example 6: Using Parentheses When Differentiating • , • , • , • ,
Theorem 3.5: The Sum & Difference Rules THEOREM: The sum (difference) of two differentiable functions f and g is itself differentiable. Moreover, the derivative of () is the sum (difference) of the derivatives of f and g. PROOF The proof is a direct consequence of theorem 2.2 (sum of limits).
Derivatives of Sine & Cosine Functions • Recall from the previous chapter that • We will use these limits to find the derivatives of the sine and cosine functions • We will also use the following identities
Derivatives of Sine & Cosine Functions THEOREM: PROOF From the definition
Derivatives of Sine & Cosine Functions THEOREM: PROOF From the definition
Example 8: Derivatives Involving Sines & Cosines • , • , • ,
Theorem 3.7: Derivative of the Natural Exponential Function THEOREM: The natural exponent function is the unique function that is equal to its own derivative. Another way to think about this: At each point on the graph of the function, the function value is the same as the slope of the tangent line at that point.
Example 9: Derivatives of Exponential Functions • , • , • ,
Rates of Change • The value of the derivative of a function f at a point on the graph is the slope of the tangent line at that point • But a slope is a rate of change of the y-values with respect to the x-values • We will often use this interpretation of the derivative when solving problems • As such, pay close attention to the units! • A common use for rate of change is to describe the motion of a particle in a straight line
Rates of Change • We usually designate this motion as horizontal or vertical (with respect to the x- and y-axes) • If horizontal, movement to the right is considered positive while movement to the left is negative • If vertical, movement up is positive and movement down is negative • A function s that gives the position of a particle (with respect to the origin) at time t is called the position function
Rates of Change • You should by now be familiar with the formula • If, over a period of time a particle changes its position by , then • Note that the numerator is a change in position (hence we would use units of length) while the denominator is a change in time (times units)
Rates of Change • So is , each with units of change in position (or distance) over change in time • Note that we use as a formula only when the rate is constant • The reason is that both and represent average velocity • When the rate (or velocity) is constant, then the average is exactly • To illustrate, suppose that you drive on a straight and level road that is 10 miles long • If at time hours your speedometer is not moving nor does it move during the entire 20 minutes (or 1/3 hour), then you average velocity is
Rates of Change • Now suppose that you drive from Del Rio to San Antonio, a distance of 150 miles, and it takes you 2.5 hours • Your average velocity is • But note that, on such a trip, your speedometer does move • That is, you were not driving 60 mph during the entire 2.5 hours • During some periods you would have been driving faster and a others slower, or even stopped
Example 10: Finding Average Velocity of a Falling Object If a billiard ball is dropped from a height of 100 feet, its height s at time t is given by where s is measured in feet and t is measured in seconds. Find the average velocity over each of the following time intervals a) [1, 2] b) [1, 1.5] c) [1, 1.1]
Example 10: Finding Average Velocity of a Falling Object For each interval , use the formula
Instantaneous Rate of Change • When velocity is constant, the graph of the position function is a straight line (slope, which is rate of change, is a constant value) • If the graph of the position function is not a straight line, then the velocity differs from one instant in time to the next • The velocity at a particular point in time is the instantaneous velocity • The instantaneous velocity is the slope of the line tangent to the graph at a given point • In other words, the instantaneous velocity at a point is given by the value of the derivative at the point
Instantaneous Rate of Change • In general, if is the position function for an object moving along a straight line, the velocity of the object at time t is • Velocity is a vector function so it can be positive, negative, or zero • Speed is the absolute value of velocity • The position function for a free-falling object is given by • Here, g is acceleration due to gravity, is the initial velocity and is the initial position
Example 11: Using the Derivative to Find Velocity At time , a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by where s is measured in feet and t is measured in seconds. • When does the diver hit the water? • What is the diver’s velocity at impact?
Example 11: Using the Derivative to Find Velocity For part a), the position function, with , will allow you to solve for t So . Only the positive value has meaning here, so the diver reaches the water after 2 seconds. The velocity at impact is the derivative of s at time
Exercise 3.2a • Page 136, #1-30
Exercise 3.2b • Page 136, #31-61 odds, 63-66, 73-76
Exercise 3.2c • Page 138, #89-100, 103-115 odds