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
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′ 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