Chapter 3 Derivatives

DERIVATIVES (DIFFERENTIATION) ZONE FOR AP CALCULUS AB

Defintion Graphical and Physical

In this note we would be taking a look at derivatives from a multi representational perspective.
The derivative of a function can be defined as the slope of a tangent line to a curve(grahical definition). It is also the instantaneous rate of change of a function(physical definition).

The Limit Definition of a Derivative
The derivative of a function at x can be found by computing the following limits

COMPLETE DERIVATIVES PLAYLIST

DERIVATIVES REVIEW TUTORIALS

1. Differentiation Rules Pt I

1b. Differentiation Rules Pt II

1c. Differentiation Rules Pt III

1d. Differentiation Rules Witch of Agnesi

2a. Application of Derivatives to Velocity, Position, Acceleration and Jerk

3. Chain Rule

4. Logarithmic Differentiation I

5. Logarithmic Differentiation II

6. Derivative of Inverse Trigonometric Functions

COLLECTION OF VIDEO TUTORIALS

CU3L1 Introduction to Derivatives Differentiation Rules AP Calculus AB BC Exam IB

Product and Chain Rule AP Calculus Derivatives Differentiation

CU3L2 Application of Derivatives Motion Acceleration Position Free fall

cu3l3a Chain Rule Differentiation Technique

chain rule differentiation pt I

chain rule differentiation pt II

Cu3L4 Implicit Differentiation pt I vertical and horizontal tangents

Cu3L4b Implicit Differentiation pt II Vertical and Horizontal Tangent lines Calculus AB BC

CU3L4c Logarithmic Differentiation pt I Calculus AB BC AP

CU3L4c Logarithmic Differentiation pt II Calculus AB BC AP IB

logarithmic differentiation calculus AB BC implicit derivative

Cu3L5 Derivative of inverse trig functions arcsecant arcsin arccos arctan arccsc Calculus AB

Chapter 3 Test Review Differentiation AP Calculus AB BC IB Exam calculus I

The Chain Rule Differentiation Calculus AB IB Exam BC Derivative Composite

Review Differentiation implicit differentiation part III cu3l7c Calculus I AP AB BC IB

Connecting f' with the graph of f applications of derivative 4 3

Differentiation Rules Derivatives AP Calculus AB

Derivatives Review Part I chapter test review Differentiation AP Calculus AB BC IB Exam

logarithmic differentiation calculus AB BC implicit derivative

DERIVATIVES

Partilhar

RELATED LINKS

LECTURE NOTES

VIDEO NOTES

DIFFERENTIATION RULES

Power Rule for Positive Integer Power of x

If n is a positive integer, then

d/dx(x^n)=nx^n-1

Example:

d/dx (x^3) =3x^(2-1)=3x^2

The Constant Multiple Rule

If u is a differentiable function of x and c is a constant, thdn

d/dx (cu) = c d/dx(u)

Example:

Find the derivative of the function y=3x^4

y=3x^4
y'=(3x^4)^4 = 3(x^4)' =3(4x^3 )=12x^3

THE CHAIN RULE
If f is differentiable at the point u =g(x), and g is differentiable at x, then the
composite function f (g(x)) is differentiable at x, and
[ f ( g(x))]' =f ' (g) • g'(x).
In Leibniz notation, if y =f( u) and u =g(x), then
dy/dx=dy/du*du/dx

Example 1

Find dy/dx for the function y=sin(cosx))

dy/dx=dy/du*du/dx

y=sin u where u cos x
so

dy/dx=(sin u)' * u'

dy/dx=cos u* u'

dy/dx=cos(cosx)(cos x)'

dy/dx=cos(cos x)(-sin x)
dy/dx=-sinxcos(cos x)

AP Calculus Standards Covered in this Section
APC.5 The student will investigate derivatives presented in graphic, numerical, and analytic
contexts and the relationship between continuity and differentiability. The derivative will
be defined as the limit of the difference quotient and interpreted as an instantaneous rate
of change.
Mathematics Standards of Learning
60
APC.6 The student will investigate the derivative at a point on a curve. This will include
a) finding the slope of a curve at a point, including points at which the tangent is vertical
and points at which there are no tangents;
b) using local linear approximation to find the slope of a tangent line to a curve at the
point;
c) defining instantaneous rate of change as the limit of average rate of change; and
d) approximating rate of change from graphs and tables of values.
APC.7 The student will analyze the derivative of a function as a function in itself. This will
include
a) comparing corresponding characteristics of the graphs of f, f ', and f ";
b) defining the relationship between the increasing and decreasing behavior of f and the
sign of f ';
c) translating verbal descriptions into equations involving derivatives and vice versa;
d) analyzing the geometric consequences of the Mean Value Theorem;
e) defining the relationship between the concavity of f and the sign of f "; and
f) identifying points of inflection as places where concavity changes and finding points
of inflection.