# Derivatives of Rational and Trigonometric Functions

## 2.1 Definition of a Derivative

We've been finding general formulas for the slope of a tangent to a curve.  This is such an important concept that it has its own name.  This is called the derivative of a function.  Today we will continue exploring ways to find the derivative of a function.

Resources:

• notes

Assignment:

• p101 #1-6all, 11, 15

## 2.2 Relating Graphs of a function and its derivative

Remember that the derivative of a function tells us the slope of the tangent line for all points on the graph of a function. This means that a graph of the derivative can be developed, as a function has an infinite number of points, and each point has its own tangent line.  Today you will see how the graph of the derivative is related to the graph of the function.

Things you need to know

• how to create a rough sketch of the derivative of a function based on the funcion
• why you can't create a sketch of a function from its derivative without a coordinate

Resources

• Notes

Assignment:

• p101 #7-10, 13, 16-19, 21

## 2.3 Differentiability

Since derivatives are derived from limits, there are some circumstances where a function is not differentiable at certain locations, much like when it is not possible to determine the limit at a certain point. Today you will explore some of the circumstances where a function is not differentiable at a point, and the reasons why it is not differentiable.

Things you should know after today:

• What are the 4 kinds of places where a function may not be differentiable
• What is local linearity?
• How can you use your graphing calculator to estimate a numerical derivative, and what are some of the limitations of doing so.
• How does the intermediate value theorem apply to derivatives>

Resources:

• Notes

Assignment:

• p111 #1-23 odds

## 2.4 Rules for Differentiation

There are several shortcuts that can be used to find the derivatives of functions. Today you will develop some of the rules that can be used to help differentiate functions in different formats.

Things you should be able to do after today:

• Use the rules for differentiation to differenitate:
• polynomials
• products of functions
• quotients of functions
• higher order derivatives

Resources

• Notes

Assignment

• p120 #1-19 odd, 20-22

## 2.5 Tangents to a Curve

One of the important reasons for being able to find the derivative is to find the equation of tangent lines.  Additionally, the normal line, which is related to the tangent line, is an important feature of many functions.  Why is it called a normal line?  I feel like there is a joke here, but I can't think of a good punch line.

Things you should be able to do after today:

• Given a function, determine the equation of a tangent line at a specific point
• Given a function, determine the equation of the normal line at a specific point
• Find the location of any points with a specific, given slope of a tangent or normal

Resources

• Notes

Assignment

• p120 #25, 27-30, 31, 33, 34, 36, 37, 39

## 2.6 Velocity and Acceleration

One of the best examples of where a function, its derivative and its second derivative are important is in problems involving displacement, velocity and acceleration.  It especially has importance in problems involving ballistic motion (sometimes called parabolic or projectile motion) or in situations that can be modeled with functions.

Things you should be able to do after today:

• Solve problems using first and second derivatives

Resources

• Notes 2.6
• Notes 2.7

Assignment

• 2.6 p129 #1-6, 12, 14, 16, 21, 23
• 2.7 p130 #8-11 25, 27, 29, 31, 33, 34

## 2.8 Derivatives of Trig Functions

Today you will be differentiating sine and cosine functions to develop the derivatives for these.  Once you have developed the derivatives for these, we can use our quotient rules and known trig identities to develop the derivatives for the remaining 4 trigonometric functions.

Things you should be able to do after today:

• find the derivatives of composite (sum and difference, product and quotient)  functions involving trigonometric functions
• Solve problems involving derivatives of trigonometric functions

Resources:

• Notes

Assignment:

• p140 #1-13 odd, 14, #15-25 odd, #29-35 odd

## 2.9 Chain Rule

We have been differentiating functions where the input of the function was x, and it is easy to differentiate with respect to x.  Sometimes, we will differentiate composite functions where the input of one function is another functinon entirely.  In these cases, we will need to use the Chain Rule.  Chain rule is a very important and very useful method of differentiation.  It is absolutely critical for doing implicit differentiation which we will explore next class.

Things you should be able to do after today:

• apply the chain rule to differentiate a composite function

Resources

• Notes

Assignment

• p140 #-13 odd, #14, #15-25 odd, #29-35 odd
• p146 #1-37 odd