Mr. Jonathan AndersonMr. Jonathan AndersonMAT – 1710 Calculus & Analytic Geometry IMAT – 1710 Calculus & Analytic Geometry I

SUNY JCCSUNY JCCJamestown, NYJamestown, NY

Vocab and NotationVocab and Notation

The process of finding the derivative of a function is called differentiation.

To differentiate is to find the derivative.

4 different notations:

′f x( ) ≡dy

dx≡

d

dxy[ ] ≡ ′y

ContinuityContinuity

Differentiability implies continuity.

Continuity does not imply differentiability (sharp point).

A function is differentiable at a point (x = c) iff a single tangent line to the curve can be drawn at that point.

Limit DefinitionLimit Definition

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d

dxf x( )[ ] = lim

h→0

f x + h( ) − f x( )

h

ExampleExample

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f x( ) = x 3

€

′ f x( ) = limh→0

f x + h( ) − f x( )

h

= limh→0

x + h( )3

− x 3

h

= limh→0

x 3 + 3hx 2 + 3h2x + h3( ) − x 3

h

= limh→0

3hx 2

h+

3h2x

h+

h3

h

= 3x 2.

Constant RuleConstant Rule

The derivative of a constant is zero.

Example:

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f x( ) = 4 ⇒ ′ f x( ) = 0

Power RulePower Rule

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If f x( ) = g x( )[ ]n, then ′ f x( ) = n ⋅ g x( )[ ]

n −1⋅ ′ g x( )

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Examples :

f x( ) = x 3 ⇒ ′ f x( ) = 3x 2

g x( ) = 3 x − 2( )5 ⇒ ′ g x( ) =

Sum / DifferenceSum / Difference

The derivative of a sum/difference is the sum/difference of the derivatives.

f x( ) =3x2 + 2x−1dydx

=ddx

3x2⎡⎣ ⎤⎦+ddx

2x[ ] +ddx

−1[ ]

=6x+ 2

Sine and CosineSine and Cosine

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d

dxsin x[ ] = cos x

d

dxcos x[ ] = −sin x

The Exponential The Exponential FunctionFunction

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d

dxex

[ ] = ex

Application to SlopeApplication to Slope

If you evaluate the first derivative at x=c, that will be the slope of f(x) at x=c.

Therefore, the first derivative of a given function is the function that is collection of points representing the slopes of the given function.

ExampleExample

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Find the slope of f x( ) = x 3 at 2, 8( ).

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f x( ) = x 3 ⇒ ′ f x( ) = 3x 2

′ f 2( ) = 3 2( )2

= 12

⇒ The slope of f x( ) at 2, 8( ) is 12.

HomeworkHomework

Pg. 136 # 3-23 [5], 33-49 EOO, 56, 59-61