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Limits Pre-Calculus Calculus

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**The Derivative Objectives: Discuss slope and tangent lines.**

Be able to define a derivative. Be able to find the derivative of various functions. Critical Vocabulary: Slope, Tangent Line, Derivative

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I. Slopes of Graphs What are the slopes of the following linear functions? How have you defined slope in the past?

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I. Slopes of Graphs What about other functions? In these functions the slopes vary from point to point

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I. Slopes of Graphs To find the rate of change (slope) at a single point on the function, we can find a tangent line at that point Recall Circles: A line was tangent to a circle if it intersected the circle ONCE.

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**Is the slope the same at each of these arbitrary points?**

I. Slopes of Graphs To find the rate of change (slope) at a single point on the function, we can find a tangent line at that point Even though the tangent line is touching the graph someplace else, we are only describing the slope at the point of tangency. Is the slope the same at each of these arbitrary points? With Curves (functions), it is a little different. We can be concerned with the line of tangency at a specific point, even if the line would intersect the function someplace else.

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**II. Defining a Derivative**

We will be using the idea of limits to help us define a tangent line. Important things to know: 1. (x, y) is the same as (x, f(x)) 2. What does Δx represent? Change in x-values Let’s look at the slope formula: Define our points: A: (x, f(x)) B: (x + Δx, f(x + Δx)) Find the slope between points A and B This is the Difference Quotient

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**II. Defining a Derivative**

The slope of the tangent line at any given point on a function is called a derivative of the function and is defined by:

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**III. Finding the Derivative**

Example 1: Find the derivative of f(x) = -2x + 4 using the definition of the derivative. This is the general rule to find the slope at any given point on the graph.

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**What is the slope of the tangent line at the point (-1, 2)? (2, 5)?**

III. Finding the Derivative Example 2: Find the derivative of f(x) = x2 + 1 using the definition of the derivative. What is the slope of the tangent line at the point (-1, 2)? (2, 5)? (-1, 2): Slope would be -2 This is the general rule to find the slope at any given point on the graph. (2, 5): Slope would be 4

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**Big Ideas III. Finding the Derivative**

2x was the DERIVATIVE of f(x) = x This means it is the general rule for finding the slope of the tangent line to any point (x, f(x)) on the graph of f. We write this by saying f’(x) = 2x. We say this “f prime of x is 2x” 3. The process of finding derivatives is called DIFFERENTIATION Notations:

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**III. Finding the Derivative**

Example 3: Find the derivative of f(x) = 3x2 – 2x using the definition of the derivative. This is the general rule to find the slope at any given point on the graph.

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**What is the slope of the tangent line at (2, 1)?**

III. Finding the Derivative Example 4: Find the slope of g(x) = 5 - x2 at (2, 1) What is the slope of the tangent line at (2, 1)? (2, 1): Slope would be -4 This is the general rule to find the slope at any given point on the graph.

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**What is the slope of the tangent line at (-3, 4)?**

III. Finding the Derivative Example 5: Find the equation of the tangent line to the graph of f(x) = x2 + 2x + 1 at the point (-3, 4). What is the slope of the tangent line at (-3, 4)? -4 y = mx + b 4 = (-4)(-3) + b f(x) = -4x - 8 4 = 12 + b -8 = b

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Implicit Differentiation Objective: To find derivatives of functions that we cannot solve for y.

Implicit Differentiation Objective: To find derivatives of functions that we cannot solve for y.

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