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An introduction to the difference quotient, a fundamental concept in calculus. It explains the setup and simplification of difference quotients, using examples with various functions including polynomials, radical functions, and rational functions. Understanding difference quotients is essential for developing derivative formulas.
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2
2
2
2
2
2
2
2 2
2
2 2
2 2
2
2 3(25 + 10h + h ) − 20 − 4h − 5
2
= 75 + 30h + 3h − 20 − 4h − 5
2
= 3h + 26h + 50
f(x h) f(x)
h
f(a h) f(a)
h
f(5 h) f(5)
h
f(x x) f(x)
x
2
f(x h) f(x)
h
2 2 2
[3x 6xh 3h 4x 4h 5] (3x 4x 5)
h
2 2 2
3x 6xh 3h 4x 4h 5 3x 4x 5
h
2
6xh 3h 4h
h
h(6x 3h 4)
h
= (^) 6x + 3h − 4
f(a h) f(a)
h
6a + 3h − 4
f(5 h) f(5)
h
2 2
2 [3(25 10h h ) 20 4h 5] (75 20 5)
h
2 [3h 26h 50] (50)
h
2 3h 26h h(3h 26)
3h 26
h h
f(x h) f(x)
h
h
x h x x h x
h x h x
(^) + − (^) + +
(x h) x h 1
h( x h x ) h( x h x ) x h x
f(3 h) f(3)
h
h
h (^3) h 3
(3 h) 3 h 1
h( 3 h 3) h( 3 h 3) 3 h 3
4x
x − 5
f(x h) f(x)
h
4(x h) 4x
x h 5 x 5
h
+
−
(x h 5)(x 5)
(x h 5)(x 5)
+ − −
4(x h)(x 5) 4x(x h 5)
h(x h 5)(x 5)
− − + −
− −
2 2 4x 20x 4xh 20x 4x 4xh 20x
h(x h 5)(x 5)
− + − − − −
20h 20
h(x h 5)(x 5) (x h 5)(x 5)
− −
=
f(x h) f(x)
h
2 − 5x + 3x − 7
3 4x + 6
3x
4 −2x
4 2x
3x 1
− 10x − 5h + 3
2 2 12x + 12xh +4h
7
7x + 7h − 8 + 7x − 8
5
9 5x 5h 9 5x
−
− − + −
3
(2 − x)(2 − x −h)
14
(3x 1)(3x 3h 1)
−