3.1 – Derivative of a Function

Slope of the Tangent Line If f is defined on an open interval containing c and the limit exists, then and the line through (c, f (c)) with slope m is the line tangent to the graph of f at the point (c, f (c)).

The Slope of the Graph of a Linear Function Find the slope of the graph of at the point (2, 1).

The Slope of the Graph of a Nonlinear Function Find the slope of the graph of at the point (0, 1) and (-1, 2). Find the equation of the tangent line at each point.

Definition Derivative – The derivative of f at x is given by provided the limit exists. For all x for which this limit exists, is a function of x.

Find the Derivative of the Function

Alternate Definition The derivative of the function f at the point x = a is the limit provided the limit exists.

Find the Derivative of the Functions

Homework p.105 ~ 1-9 (O), 13-16, 17, 19

Reflection p.105 ~ 1-9 (O), 13-16, 17, 19

Differentiation Rules 3.3.1 (also 3.2)

When Derivatives Do Not Exist

When Derivatives Do Not Exist

When Derivatives Do Not Exist

When Derivatives Do Not Exist

The Constant Rule The derivative of a constant function is 0. That is, if c is a real number, then .

The Power Rule If n is a rational number, then the function f (x) = xn is differentiable and For f to be differentiable at x = 0, n must be a number such that xn–1 is defined on an interval containing 0.

Find the Derivative of the Function

The Constant Multiple Rule If f is a differentiable function and c is a real number, then cf is also differentiable and .

Find the Derivative of the Function

The Sum and Difference Rules The sum (or difference) of two differentiable functions f and g is itself differentiable. Moreover, the derivative of f + g (or f – g) is the sum (or difference) of the derivatives of f and g.

Find the Derivative of the Function

The Slope of a Graph Find the slope of the graph of when , , and .

The Tangent Line Find an equation of the tangent line to the graph of when .

The Product Rule The product of two differentiable functions f and g is itself differentiable. Moreover, the derivative of fg is the first function times the derivative of the second, plus the second function time the derivative of the first.

Find the Derivative

Find the Derivative

The Quotient Rule The quotient of two differentiable functions f and g is itself differentiable for all values of x for which g(x) ≠ 0. Moreover, the derivative of f / g is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Find the Derivative

Find the Derivative

Homework p. 114 ~ 1-10 p. 124 ~ 1-5, 7-29 (O), 30, 32

Higher Order Derivatives 3.3.2

Which Rule Do I Use? Find the Derivative

Higher-order Derivative Notation First Derivative: Second Derivative: Third Derivative: Fourth Derivative: nth Derivative:

Find the Second Derivative

Instantaneous Rate of Change A population of 500 bacteria is introduced into a culture and grows in number according to the equation where t is measured in hours. Find the rate at which the population is growing when t = 2

Velocity and Other Rates of Change 3.4

Position and Velocity If a billiard ball is dropped from a height of 100 feet, its height s at time t is given by the position function where s is measured in feet and t is measured in seconds. Find the average velocity over each of the following time intervals. [1, 2] [1, 1.5] [1, 1.1]

Position Function

Instantaneous Velocity At time t = 0, 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?

Higher-order Derivatives Because the moon has no atmosphere, a falling object on the moon encounters no air resistance. In 1971, astronaut David Scott demonstrated that a feather and a hammer fall at the same rate on the moon. The position function for each of these falling objects is given by where s(t) is the height in meters and t is the time in seconds. What is the ratio of Earth’s gravitational force to the moon’s?

Free Fall A silver dollar is dropped from the top of a building that is 1362 feet tall. Determine the position, velocity, and acceleration functions for the coin. Determine the average velocity on the interval [1, 2]. Find the instantaneous velocities when t = 1 and t = 2. Find the time required for the coin to reach ground level. Find the velocity of the coin at impact.

Free-Fall A projectile is shot upward from the surface of Earth with an initial velocity of 120 meters per second. What is its velocity after 5 seconds? What is the maximum height of the projectile?

Homework p.135/1, 3, 7, 9-15 (O), 19, 21

Derivatives of Trigonometric Functions 3.5

The Derivative of Sine

Derivatives of Sine and Cosine Functions

Find the Derivative of the Function

Find the Derivative

Simple Harmonic Motion A weight hanging from a spring is stretched 5 units beyond its rest position (x = 0) and released at time t = 0 to bob up and down. Its position at any later time t is What are its velocity and acceleration at time t? Describe its motion.

Derivative of Tangent

Derivatives of Trigonometric Functions

Find the Derivative

Find the Second Derivative

Homework p. 146/ 1-9 (O), 11-23 (O), 27-39 (O), 43

The Chain Rule 4.1

The Chain Rule If y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f (g(x)) is a differentiable function of x and or, equivalently,

Identify the inner and outer functions Composite y = f (g(x)) Inner u = g(x) Outer y = f (u) 1. 2. 3. 4.

The General Power Rule If , where u is a differentiable function of x and n is a rational number, then or, equivalently,

Find the Derivative

Find the Derivative

Homework Chain Rule Worksheet

Factoring Out the Least Powers Find the Derivative

Factoring Out the Least Powers Find the Derivative

Factoring Out the Least Powers Find the Derivative

Find the Derivative

Find the Derivative

Trig Tangent Line Find an equation of the tangent line to the graph of at the point (π, 1). Then determine all values of x in the interval (0, 2π) at which the graph of f has a horizontal tangent.

Homework p.153/ 1-11odd, 21-39odd, 59

Implicit Differentiation 4.2

Find dy/dx

Guidelines for Implicit Differentiation Differentiate both sides of the equation with respect to x. Collect all terms involving dy / dx on the left side of the equation and move all other terms to the right side of the equation. Factor dy / dx out of the left side of the equation. Solve for dy / dx.

Find the derivative

Homework p.162/ 1-19odd, 49, 51

Example Determine the slope of the tangent line to the graph of at the point .

Example Determine the slope of the tangent line to the graph of at the point .

Finding the Second Derivative Implicitly

Finding the Second Derivative Implicitly

Example Find the tangent and normal line to the graph given by at the point .

Homework p.162/ 21-25odd, 27-30, 31-43odd

Inverse Functions 4.3

Definition of Inverse Function A function g is the inverse function of the function f if for each x in the domain of g. and for each x in the domain of f.

Verifying Inverse Functions Show that the functions are inverse functions of each other. and

The Existence of an Inverse Function A function has an inverse function if and only if it is one-to-one. If f is strictly monotonic on its entire domain, then it is one-to-one and therefore has an inverse function.

Existence of an Inverse Function Which of the functions has an inverse function?

Finding an Inverse Find the inverse function of .

The Derivative of an Inverse Function Let f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which . Moreover,

Example Let . What is the value of when x = 3?

Homework p. 44/ 1-6, 7-23odd, 43 p. 170/ 28, 29bc

Inverse Trigonometric Functions 3.8

The Inverse Trigonometric Functions

Evaluating Inverse Trigonometric Functions Evaluate each function.

Solving an Equation

Using Right Triangles a) Given y = arcsin x, where , find cos y. b) Given , find tan y.

Homework 3.8 Inverse Trig Review worksheet

Derivatives of Inverse Trigonometric Functions Find

Derivatives of Inverse Trigonometric Functions

Differentiating Inverse Trigonometric Functions

Differentiate and Simplify

Homework p. 170/ 1-27odd, 31ab

Derivatives of Exponential and Logarithmic Functions 4.4

Properties of Logarithms If a and b are positive numbers and n is rational, then 1. 2. 3. 4.

Expand the following logarithms

Solving Equations Solve 7 = ex + 1 Solve ln(2x  3) = 5

Derivative of the Natural Exponential Function

Examples

Derivatives for Bases Other than e

Find the derivative of each function.

Homework p.44/ 33-38 p.178/ 1-13odd, 29, 30

Derivative of the Natural Logarithmic Function

Example Find the derivative of ln (2x).

Find the Derivative 1. 2. 3.

Derivatives for Bases Other than e

Example Differentiate .

Example Differentiate .

Example Differentiate .

Find an equation of the tangent line to the graph at the given point.

Homework p. 178/15-27odd, 37-41odd

Comparing Variable and Constants

Derivative Involving Absolute Value

Find the derivative

Use implicit differentiation to find dy/dx. ln xy + 5x = 30

Logarithmic Differentiation Differentiate .

Logarithmic Differentiation

More Logarithmic Differentiation

Homework p. 179/ 31, 33-36, 43-55odd