[the quotient rule][section 6]

(1.) f(t) = 1/t6               here is the problem

f(t) = t-6                use a negative exponent

f'(t) = -6t-7           use the power rule

f'(t) = -6/t7            simplify

(2.)  g(x) = 5/(7x3)           here is the problem

g(x) = (5/7)x-3          use a negative exponent

g'(x) = (-15/7)x-4    use the power rule

g'(x) = (-15)/(7x4)       simplify

(3.)  g(t) = 2(t2 + 1)/t            here is the problem

g(t) = 2t + 2t-1    multiply thru and divide thru by t

g'(t) = 2 - 2t-2       use the power rule

g'(t) = 2 - (2/t2)       simplify

(4.)  G(x) = (1 + x + x5)/(2 - x + x6)

G'(x) = (5x4 + 1)(2 - x + x6) - (-1 + 6x5)(1 + x + x5)
____________________________________________
(2 - x + x6)2

[use the quotient rule]

_
(5.)  A(x) = 1/√x           here is the problem

A(x) = x-1/2          use a negative rational exponent

A'(x) = (-1/2)x-3/2          use the power rule

A'(x) = (-1)/(2x3/2)      simplify
_
(6.)  h(z) = z3/(1 + √z)        here is the problem

h'(z) =  (3z2)[1 + z1/2] - (z3)[(1/2)z-1/2]  use the quotient
_________________________________ rule
(1 + z1/2)2
_
(7.)  g(t) = t2/(1 + √t)  ;   (4, 16/3)  here is the problem

g'(t) = [1 + t1/2](2t) - [(1/2)t-1/2](t2)  use the quotient
______________________________ rule
[1 + t1/2]2

g'(t) =  2t + t3/2 - (1/2)t3/2
______________________    multiply thru on top
[1 + t1/2]2

g'(4) =  2(4) + 43/2 - (1/2)(4)3/2
_________________________   replace t with 4
[1 + 41/2]2

g'(4) =  (8 + 8 - 4)/(1 + 2)2    simplify

g'(4) = 12/9              combine like terms and sqr 3

g'(4) = 4/3              reduce

y - y1 = m(x - x1) use this line formula

y - (16/3) = (4/3)(x - 4)  make substitutions

3y - 16 = 4x - 16     multiply thru parentheses

+16    +  16    add 16 to each side
_______________________

result:  3y = 4x

(8.)  h(u) = (1 + √u)/(u2)  (1,2)

h(u) = u-2 + u-3/2           divide thru by u2

h'(u) = -2u-3 - (3/2)u-5/2     use the power rule

h'(1) = -2(1)-3 - (3/2)(1)-5/2    replace u with 1

h'(1) = -2 - (3/2)         multiply

h'(1) = -7/2            combine like terms

y - y1 = m(x - x1)  use this line formula

y - 2 = (-7/2)(x - 1)    make substitutions

2y - 4 = -7x + 7   multiply thru by 2, multiply thru

+  4        + 4   add 4 to each side
______________________
2y      = -7x + 11         add

+7x           + 7x        add 7x to each side
_________________________
7x + 2y =            11     add

result: 7x + 2y = 11

(9.)   g(x) = 1/x7  (1,1)           here is the problem

g(x) = x-7           use a negative exponent

g'(x) = -7x-8            use the power rule

g'(1) = -7(1)-8         replace x with 1

g'(1) = -7                multiply

y - y1 = m(x - x1)  use this line formula

y - 1 = -7(x - 1)      make substitutions

y - 1 = -7x + 7     multiply thru parentheses

+  1         + 1      add 1 to each side
__________________________

y   =  -7x + 8       add

result:  y = -7x + 8

(10.)  p(x) = (x3 + 2)/(x2 + 2) ;  (0,1)

p'(x) = (3x2)(x2 + 2) - (2x)(x3 + 2)  use the quotient
____________________________ rule
(x2 + 2)2

p'(x) =   3x4 + 6x2 - 2x4 - 4x   multiply thru
_____________________ parentheses
(x2 + 2)2

p'(x) =   (x4 + 6x2 - 4x)/(x2 + 2)2    combine like terms

p'(0) = 0             replace x with 0 and simplify

y = mx + b           use this line formula

y = 0x + 1     replace m with 0 and b with 1

y = 1            multiply and add

result:  y = 1