Differentiation Calculus Chapter 2

The Derivative and the Tangent Line Problem Calculus 2.1

A tangent line… Circles – is perpendicular to the radial line at a point Curves Touches but does not cross? Touches or intersects at only one point? Has a slope equal to the slope of the curve at that point 3Calculus chapter 2

Slope of secant line As point Q approaches point P, the slope of the secant line approaches the slope of the tangent line.  slope of the tangent line is said to be the limit of the slope of the secant line 4Calculus chapter 2

Slope of tangent line f must be defined on an open interval containing c The limit must exist 5Calculus chapter 2

Linear functions Have the same slope at any point Slope of tangent line agrees with slope of line using “rise over run” definition. 6Calculus chapter 2

Example Find the slope of the graph of the following function at any point. 7Calculus chapter 2

Nonlinear functions The slope of the tangent line changes 8Calculus chapter 2

Example Find the slope of the graph of the following function at any point. 9Calculus chapter 2

Vertical tangent lines If f is continuous on [a, b], and c is in the interval [a, b], each of the following gives a vertical tangent line 10Calculus chapter 2

Definition of the Derivative of a Function The limit must exist 11Calculus chapter 2

Denoting the derivative 12Calculus chapter 2

Example Find the derivative of 13Calculus chapter 2

Example Find the slope of the graph of the following function at the points (1, 1), (4, 2), and (0,0). 14Calculus chapter 2

Alternative form of derivative Useful for finding the derivative at one point (not in general) 15Calculus chapter 2

Example Find the derivative at c = 1 16Calculus chapter 2

Differentiability implies continuity If a function is differentiable at c, then it is continuous at c The reverse is not true A continuous function might have A sharp point A vertical tangent line See page Calculus chapter 2

Basic Differentiation Rules and Rates of Change Calculus 2.2

Calculus chapter 219 Differentiation Rules Allow you to find derivatives without the direct use of the limit definition. Can be proven using the limit definition.

Calculus chapter 220 Constant Rule The derivative of a constant function is 0. The slope of a horizontal line is 0.

Calculus chapter 221 The Power Rule

Calculus chapter 222 The Constant Multiple Rule

Calculus chapter 223 The Sum and Difference Rules

Calculus chapter 224 Examples Find the derivatives

Calculus chapter 225 Derivatives of sine and cosine functions Can be proven using the special trig limits

Calculus chapter 226 Examples Find the derivatives

Calculus chapter 227 Nonexample Can’t pull the two out of the sine function A composite function Need the chain rule – section 2.4

Calculus chapter 228 Rates of change Derivatives can determine the rate of change of one variable with respect to another

Calculus chapter 229 Example The area of a circle with a radius r is A =  r 2. Find the rate of change of the area with respect to r when r = 2 m.

Calculus chapter 230 Position function Gives the position of an object as a function of time

Calculus chapter 231 Average velocity

Calculus chapter 232 Instantaneous velocity Or just velocity Derivative of the position function

Calculus chapter 233 Speed Absolute value of velocity

Calculus chapter 234 Position of free-falling object s 0 is initial height v 0 is initial velocity g is acceleration due to gravity –32 ft/s 2 –9.8 m/s 2

Calculus chapter 235 Example A ball is thrown straight down from the top of a 220-ft building with an initial velocity of –22 ft/s. What is its velocity after 3 seconds? What is its velocity after falling 108 feet?

The Product and Quotient Rules and Higher-Order Derivatives Calculus 2.3

Product Rule Take the first function times the derivative of the second plus the second function times the derivative of the first. 37Calculus chapter 2

Product rule with more than two functions Take the derivative of each function times the other functions and add all products 38Calculus chapter 2

Examples Find the derivatives using the product rule. 39Calculus chapter 2

The quotient rule Take the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator divided by the square of the denominator. 40Calculus chapter 2

The quotient rule 41Calculus chapter 2

Examples Find the derivatives using the quotient rule 42Calculus chapter 2

Avoiding the quotient rule See example 6 on page Calculus chapter 2

Derivatives of trig functions Can be proven using the quotient rule 44Calculus chapter 2

Higher order derivatives Second derivative of the function is the derivative of the first derivative Third derivative of the function is the derivative of the second derivative Etc. See page Calculus chapter 2

Acceleration The derivative of velocity. The second derivative of position. Example: find the acceleration of an object in free fall if the position function is 46Calculus chapter 2

The Chain Rule Calculus 2.4

The Chain rule One of the most powerful differentiation rules Used for composite functions 48Calculus chapter 2

The Chain Rule Differentiate the “outside” function, leaving the inside function alone Then multiply it by the derivative of the inside function Example: 49Calculus chapter 2

Examples Differentiate 50Calculus chapter 2

General Power Rule 51Calculus chapter 2

Examples Differentiate 52Calculus chapter 2

Examples Differentiate 53Calculus chapter 2

Implicit Differentiation Calculus 2.5

Calculus chapter 255 Explicit functions y is an explicit function of x Solved for y, or easy to solve for y

Calculus chapter 256 Implicit functions Can’t be easily solved for y

Calculus chapter 257 Implicit differentiation Used to differentiate implicit functions To differentiate with respect to x For each term that involves x alone, proceed as normal For each term that involves y, apply the chain rule

Calculus chapter 258 Examples

Calculus chapter 259 Guidelines 1.Differentiate both sides with respect to x 2.Collect all terms with dy/dx on the left and all other terms on the right. 3.Factor out dy/dx. 4.Solve for dy/dx.

Calculus chapter 260 Examples

Calculus chapter 261 Representing a Graph by differentiable functions

Calculus chapter 262 Example Find the slope of the tangent line to the graph at the indicated point

Calculus chapter 263 Finding the second derivative Find the second derivative with respect to x in terms of x and y.

Related rates Calculus 2.6

Finding related rates Use the chain rule to find the rates of change of two or more related variables that are changing with respect to time. 65Calculus chapter 2

Example Differentiate with respect to t Insert known values 66Calculus chapter 2

Example The radius r of a sphere is increasing at a rate of 2 inches per minute. Find the rate of change of the volume of the sphere when r = 6 inches. 67Calculus chapter 2

Example A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at the rate of 10 cubic feet per minute, find the rate of change of the depth of the water the instant it is 8 feet deep. 68Calculus chapter 2

More examples Pages 153 – Calculus chapter 2