Announcements Topics: -sections 4.4 (continuity), 4.5 (definition of the derivative) and (differentiation rules) * Read these sections and study solved examples in your textbook! Work On: -Practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the link “SCHEDULE + HOMEWORK”)

The Derivative Definition: Given a function f(x), the derivative of f with respect to x is the function f’(x) defined by The domain of this function is the set of all x- values for which the limit exists.

The Derivative Definition: Given a function f(x), the derivative of f with respect to x is the function f’(x) defined by The domain of this function is the set of all x- values for which the limit exists.

The Derivative Interpretations of f’: 1.The function f’(x) tells us the instantaneous rate of change of f(x) with respect to x for all x-values in the domain of f’(x). 2. The function f’(x) tells us the slope of the tangent to the graph of f(x) at every point (x, f(x)), provided x is in the domain of f’(x).

The Derivative Example: Find the derivative of and use it to calculate the instantaneous rate of change of f(x) at x=1. Sketch the curve f(x) and the tangent to the curve at (1,2).

The Derivative Example: Find the derivative of Sketch the graph of f(x) and the graph of f’(x).

Relationship between f’ and f If f is increasing on an interval (c,d): The derivative f’ is positive on (c,d). The rate of change of f is positive for all x in (c,d). The slope of the tangent is positive for all x in (c,d). If f is decreasing on an interval (c,d): The derivative f’ is negative on (c,d). The rate of change of f is negative for all x in (c,d). The slope of the tangent is negative for all x in (c,d).

Critical Numbers Definition: c is a critical number of f if c is in the domain of f and either f’(c)=0 or f’(c) D.N.E.

Differentiable Functions A function f(x) is said to be differentiable at x=a if we are able to calculate the derivative of the function at that point, i.e., f(x) is differentiable at x=a if exists.

Differentiable Functions Geometrically, a function is differentiable at a point if its graph has a unique tangent line with a well- defined slope at that point. 3 Ways a Function Can Fail to be Differentiable:

Graphs Example: (a)Sketch the graph of (b) By looking at the graph of f, sketch the graph of f’(x).

Relationship Between Differentiability and Continuity If f is differentiable at a, then f is continuous at a.

Basic Differentiation Rules All rules are proved using the definition of the derivative: The derivative exists (i.e. a function is differentiable) at all values of x for which this limit exists.

The Constant Function Rule If where is a constant, then Example:

The Power Rule If where then Example: Differentiate the following. (a)(b) (c)(d)

The Constant Multiple Rule Let be a constant. Then Example: Find the derivative of each. (a)(b)

The Sum/Difference Rule provided and are differentiable functions. Examples: Differentiate. (a) (b)

The Product Rule provided and are differentiable functions. Example: Find where

The Quotient Rule provided and are differentiable and Example: Determine where the graph of the function has horizontal tangents.

Using the Derivative to Sketch the Graph of a Function Example: Sketch the graph of

Chain Rule “derivative of the outer function evaluated at the inner function times the derivative of the inner function” Example: Differentiate the following. (a) (b)

Chain Rule Example: Using implicit differentiation, determine for

Chain Rule Example: The number of mosquitoes (M) that end up in a room is a function of how far the window is open (W, in square centimetres) according to The number of bites (B) depends on the number of mosquitoes according to Find the derivative of B as a function of W.

Derivative of the Natural Exponential Function Definition: The number e is the number for which Natural Exponential Function:

Derivative of the Natural Exponential Function Note: This definition states that the slope of the tangent to the curve at (0,1) is exactly 1, i.e.

Derivative of the Natural Exponential Function If then Proof: Note: The slope of a tangent line to the curve is equal to the value of the function at that point. Note: The slope of a tangent line to the curve is equal to the value of the function at that point.

Derivatives of Exponential Functions If then Example: Differentiate. (a) (b) (c)

Derivatives of Logarithmic Functions If then Example 1: Differentiate. (a)(b) Example 2: Determine the equation of the tangent line to the curve at the point

Derivatives of Trigonometric Functions

Example: Find the derivative of each. (a)(b) (c)

Derivatives of Inverse Trig Functions Example 1: Differentiate. (a) Example 2: Prove