Using Definition 3.6, find the derivatives of the following functions with respect to x.
1 f (x) = x 2 f (x) = 2x − 5 3 f (x) = x
2
+ 4x − 5
4 f (x) = 1
; 0 x
x
≠ 5 f (x) = x 6 f (x) = x
3
− 3
7 f (x) = (3x − 2)2 8 f (x) = x
2
(2 + x) 9 f (x) = 8 −
3
x
10 f (x) =
+ 2 3
;
3 2 2
x
x
x
≠
−
11
2
3
f x x x ( ) ; 0
x
= − ≠
12
2 3
2
4 5 ( ) ; 0 x x f x x
x
−
= ≠ 13
2
5
( ) 2 5
7
x
f x x x = − + +
14
3
2
1
f x x x ( ) ; 0
x
= + ≠ 15 3 1
f x x x x ( ) ; 0
x
= + − >
1) f'(x) = 2x
2) f'(x) = 2x + 4
3) f'(x) = -1/x^2; x ≠ 0
4) f'(x) = -1/x^2; x ≠ 5
5) f'(x) = 1/x^2; x ≠ 0
6) f'(x) = 1/x^2
7) f'(x) = 2(3x-2)(3)
8) f'(x) = x(2+x)' + (x^2)(2+x)'
f'(x) = x(1) + 2x + x^2(0)
f'(x) = 3x + 2x^2
9) f'(x) = -8/x^2
10) f'(x) = (2x^2 + x - 6)/(3x^2)
11) f''(x) = 6x; f'''(0) = 0
12) f''(x) = -24x; f'''(0) = 24
13) f'''(x) = 70/x^3
14) f''(x) = 10x; f'''(0) = 10
15) f''(x) = 6x; f'''(0) = 6