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MCV4U1UNIT 2: DERIVATIVES
Date Day Topic Homework
1 2.1 The Derivatives of Polynomial Functions Pg.73-75 #1-3,6,7,11,14,20
2 2.2 The Derivative of a Polynomial Functions Pg.82-84 # 5,11,12,18,25
3 2.3 The Product Rule Pg. 90-91 #1,2,4,5,6-8,10
4 2.4 The Quotient Rule Pg.97-98 # 2,3,6-8,9,4,5
Quiz + Work Period
5,6 2.5 The Derivatives of Composite Functions
Pg.105-106 # 4,5,6-9,15,17-19
7 Implicit Differentiation Pg. 564 #2, 3b, 5, 7, 13, 14
8 Chapter Review Pg.92-93 #1-14Pg. 110-113 #3-13, 17, 19, 22, 23, 27
9 Chapter Test
Learning GoalsIn this unit, we will learn:
Derivative (function) of a function Power Rule, Product Rule, Quotient Rule, Chain Rule To simplify derivatives
Page 1 of 20
Day 1: 2.1 Introduction to DerivativesIn the previous sections, we looked at rates of change and were able to determine the slope of a tangent line as the limit of the slopes of secant lines. This amazing result allowed us to calculate instantaneous rates of change of a function at any point. The instantaneous rate of a function is called the derivative and the study of derivatives is called differential calculus.
Does this definition look familiar? It does since it is the same formula we used in the previous chapter to find the slope of the tangent line; which is what the derivative is.
Notation: The derivative of a function y = f(x) can be denoted by the following symbols:
y ' f ' ( x )Dx ydydx
ddxf ( x )(Leibniz notation)
Derivative of f(x) at x = 3 can be written as f’(3)
Ex1: Find the derivative function of f (x)=x2 –3 x+5. Use the derivative to find the slope of the tangent atx=3 and then find the equation of the tangent.
Page 2 of 20
Definition: DerivativeThe derivative of a function w.r.t. (with respect to) the variable is the function f’ whose value at x is:
provided the limit exits [ is pronounced “f prime of x” ]
4 examples of how does not exist
1. A corner, where 1 sided derivatives differ, example
2. A cusp, where the slopes of the secants approach from one side
and - from the other side, example
3. A vertical tangent, where the slopes of the tangent
approach or - from both sides, example
4. A discontinuity, where one or both 1-sided limits FAIL to
exist, example
What do you notice about the above examples?
Using the difference of quotient to find the derivatives Page 3 of 20
Conditions for the Derivative to Exist The domain of is the set of all points for which the domain of for which the limit exists. It may be
smaller than the domain of .
If exists, we say has a derivative (is differentiable) at x. A function that is differentiable at every point of its domain is a differentiable function.
1. f (x)=x 2. f (x)=x2
3. f (x)=x3 4. f (x)=x4
5. f (x)=√x 6. f (x)=1x
Summary:yy'Day 2: 2.2 The Derivatives of Polynomial
FunctionsPage 4 of 20
So this is the lesson where most calculus students get upset because they learn that they were doing derivatives the long way. Anyway, it is important to understand where the derivative comes from, how it is defined in terms of limits, and when it is defined. Now that you completely understand the theory we can now turn our attention to the practice of calculating derivatives. So here we go.
Proof of Rule 1This should make complete sense since the function is a horizontal straight line, hence its slope should be zero.
Example of Rule 1What is the derivative of ? f(x) = 3455624?
Proof of Rule 2 (see page 77 in the textbook)
Page 5 of 20
Rule 1 Derivative of a Constant function
If is the function with the constant value , then f’(c) = 0 or
Rule 2 Power Rule for Positive Integer Powers of x
If is a positive integer, then
Example of Rule 2
What is the derivative of ?
Proof of Rule 3 (see page 78 in the textbook)
Example of Rule 3What is the derivative of ?
Page 6 of 20
Rule 3 The Constant Multiple RuleIf g(x) is a differentiable function of and k is a constant, then ddx (kg ( x ) )=k ddx [ g ( x )]
Or if f (x)=kg(x ), then f ’ (x )=kg ’(x)
Rule 4 The Sum and Difference RuleIf and are differentiable functions of , then their sum and difference are differentiable at every point where and are differentiable. At such points
ddx ( f ( x )+g ( x ) )=dfdx +
dgdx
=f ' (x )+g '(x )
Proof of Rule 4 (see page 79 in the textbook)
Example of Rule 4What is the derivative of ?
Page 7 of 20
Day 3: 2.3 The Product Rule
Proof of Rule 5
Example of Rule 5What is the derivative of ?
Using Rule 5 Expanding, then Rule 4
Example 1: Find y’.
a. Page 8 of 20
Rule 5 The Product RuleThe product of two differentiable functions f(x) and g(x) is differentiable:ddx ( f ( x )g (x ) )=f ' ( x )g ( x )+ f ( x )g '(x )
b.
c.
Example 2: Find the g’(x) if g(x) = (x + 3)(x2 – 1)(x3 + 2)
INVESTIGATION: Find y’ using the product rule for (a) and extended product rule for (b)
a . y=[g (x)]2 b . y=[g (x)]3
If y=¿then the derivative will be ____________________________.
Example 3: Find y’.
a. y=(3 x+5)10 b. y=(3 x2−2x+√ x)7
Page 9 of 20
Rule 5b The Extended Product RuleThe product of two differentiable functions and is differentiable:
ddx
(uvw )=dudxvw+u dv
dxw+uv dw
dx
Day 4: 2.4 The Quotient Rule
Proof of Rule 6 (see page 85-86 in the textbook)
Example of Rule 6
a. What is the derivative of ?
Page 10 of 20
Rule 6 The Quotient Rule
At a point where , the quotient of the two differentiable functions is
differentiable, and OR ( f (x )g(x) )
'
=f ' (x) g(x)−f ( x)g ' (x)
[ g(x )]2
b.
c.
d.
e. f. g. 34 82x xyx
2
9 3 1
yx x
y=5 x4
7
Page 11 of 20
Example of Rule 7
What is the derivative of ? Using Rule 7 Using Quotient Rule
What do you notice about this rule?
Page 12 of 20
Rule 7 Power Rule for Negative Integer Powers of x.
If is a negative integer and , then
Day 5/6: 2.5 The Chain RuleThe Derivative of Composite Functions
Using the Chain Rule (“ proper ” way using Leibniz notation)
Page 13 of 20
Rule 8 The Chain RuleIf f is differentiable at the point , and is differentiable at x, then the composite function is differentiable at x, and
( f ∘ g )' ( x )=f ' (g ( x ) )g ' (x)
In Leibniz notation, if and , then
where is evaluated at
Ex: Use the chain rule, in Leibniz notation, to find at the given value of x:
Ex: Find the derivatives of the following:
a.
b.
Page 14 of 20
Find the derivative and simplify
a. p ( x )=(2 x−5)3(3 x2+4)5
b. g ( x )=8 x3(4 x2+2 x−3)5
c. y=(5+x )2(4−7 x3)6
d. h ( x )= 6 x−1(3 x+5)4
Page 15 of 20
e. y=(2 x2−5)3
(x+8)2
f. p ( x )= −3 x4
√4 x−9
g. r ( x )=( 2 x+56−x2 )
4
h. t ( x )=[ 1( 4 x+x2)3 ]
3
Page 16 of 20
Day 7: Implicit DifferentiationConsider the following 2 equations describing familiar curves:
parabolacircle
The first equation defines y as a function of x explicitly, since for each x, the equation gives an explicit formula for finding the corresponding value of y.
The second equation does not define a function, since it fails the vertical line test. However you
can solve for y and find at least 2 functions ( and ) that are defined implicitly by the equation . If we wanted to find the derivative at a point then we would have to calculate for each implicitly defined function. In the case of the circle, there are 2 slopes for every where the derivative exists (in this example x=2 would give a vertical slope, so no derivative there).
Example at x = 1
Bottom Half Top Half
Find the equation of each tangent line:
Page 17 of 20
Here’s how we use implicit differentiation on our circle example:
Check that the points and gives the same results as when we did explicit differentiation.
Example
Find for . Then, find the slope of the tangent line at the point (2,1).
Page 18 of 20
Example
Find for . Then, find the slope of the tangent line at the point (2,-2).
Example (90o, right, perpendicular, normal, orthogonal)Show that any curve of the form for some constant c intersects any curve of the form
for some constant k, at right angles (that is the tangent lines to the curve at the intersection point are perpendicular). In this case, we say that the families of curves are orthogonal. The graph shown is when c=2 and k=5
Page 19 of 20
Day 8: Review
Derivative
Rules:
Constant Rule
Power Rule
Constant multiple Rule
Sum and different Rule
Product Rule
Quotient Rule
Chain Rule
Page 20 of 20
