CALCULUS AB
DIFFERENTIATION
BY DANIELLA KRAKUE
The Basics Behind
Differentiation
What exactly is a derivative?
The concept of Derivative is one of the
most important things you will learn in
Calculus.
When you solve for a derivative it is
called Differentiation.
 The derivative is a way to represent
rate of change, that is - the amount by
which a function is changing at one given
point.
The derivative is often written using
"dy over dx" (meaning the difference in
y divided by the difference in x).
Core Topics of Differentiation
1.
2.
3.
4.
5.
6.
7.
8.
Derivative as a Function
Product and Quotient Rules
Rates of Change
Higher Derivatives
The Chain Rule
Trigonometric Functions
Implicit Differentiation
Exponential and
Logarithmic Functions
Derivative as a Function
In order to solve for the Derivative as a Function we
use the Difference quotients. When given a
function like the one below , you plug in the given
values into the corresponding formulas and solve as
approaching the limit.
Find the derivative of the function f(x) = x2
Method 1
Method 2
More
http://www.mathopenref.com/calc
Practice! derivfunc.html
Product and Quotient Rule
 The Product Rule is expressed in this  The Quotient Rule is expressed in
formula, when given two functions
multiplied by one another you take the
derivative of the first function and
multiply it by the second then you add
the product of that to the first
function times the derivative of the
second function.
More
Practice!
the formula given. When two
functions are being divided by one
another you take the derivative of
the numerator times the
denominator, minus the derivative
of the denominator times the
numerator divided by the
denominator squared
http://www.intmath.com/differentiation/6-derivativesproducts-quotients.php
Rate of Change
The concept of Rate of Change is an application of the principles
of derivatives learned so far. ROC can best be previewed by using
this applet on the rate of a melting snowball.
http://www.mathopenref.com/calcsnowballproblem.html
More Practice:
 http://www.intmath.com/differentiation/4-
derivative-instantaneous-rate-change.php
Higher Derivatives
 The foundation for knowing how to
compute higher derivatives is not solely
knowing how to use the product/
quotient rules but also the power rule
expressed in the equation below.
Examples of Power
Rule
More Practice:
http://www.intmath.com/different
iation/9-higher-derivatives.php
The Chain Rule
 The Chain Rule is a rule that you use when you can
there is a function within a function and you can
take the derivative of the outer function.
 As shown in the example to the right the chain rule
is used with the function x2 where x = (2x + 1). The
derivative of x2 would be 2x and the derivative of
(2x + 1) =2 (x+1) and when you combine the two you
get 4(x+1).
Trigonometric Functions
 Now that the most difficult concepts
concerning derivatives has been covered on,
welcome to the derivatives of trigonometric
functions where it simply involves memorizing
the corresponding derivative to each
trigonometric function.
Practice Lesson
http://www.math.brown.edu/UTRA/trigderi
vs.html#top
Implicit Differentiation
 Implicit differentiation is
nothing more than a special case
of the well-known chain rule for
derivatives. The majority of
differentiation problems we have
discussed involve
functions y written EXPLICITLY
as functions of x. An example of
the concept defying the rules
would be …
Test you understanding
http://www.intmath.com/differentiation/8derivative-implicit-function.php
 PROBLEM 1 : Assume that y is a
function of x . Find y'
= dy/dx for x3 + y3 = 4 .
Exponential and Logarithmic
Functions
 Finally Exponential and Logarithmic
Functions . To master this concept
you simply have to remember the
following rules and apply them to
the given problems.
Derivatives of
Exponential and Logarithmic Functions
Test you understanding
http://people.hofstra.edu/stefan_waner/real
world/tutorials/unit3_3.html