Pertemuan 5
Derivatives
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Formal Definition
of the Derivative of a function
 f'(x)=

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lim
h->0
f(x+h) – f(x)
h
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Notation for derivative
 y'
 dy/dx
 df/dx
 d/dx
(f)
 f’(x)
D
(f)
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Rate of change and slope
Slope of a secant line
See diagram
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The slope of the secant line gives the
change between 2 distinct points on a
curve.
i.e. average rate of change
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Rate of change and slopeslope of the tangent line to a curve
see diagram
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The slope of the tangent line gives
the rate of change at that one point
i.e. the instantaneous change.
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compare
Slope= y-y

x-x
 Slope of secant line

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m= f ’(x)
 Slope of tangent line

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Time for examples
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
Finding the derivative
using the formal definition

This is music to my ears!
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A function has a derivative at a
point
iff the function’s right-hand and lefthand derivatives exist and are equal.
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Theorem
If f (x) has a derivative at x=c,
then f(x) is continuous at x=c.
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acceleration

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Don’t drop the ball on this
one.
13
Definition
Acceleration
The derivative of velocity,
Also ,the second derivative of
position
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Derivatives of trig functions
 Y=
sin x
 Y= cos x
 Y= tan x
 Y= csc x
 Y= sec x
 Y= cot x
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