[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]

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(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)/(2x^3/2) simplify
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(6.) h(z) = z³/(1 + √z) here is the problem

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

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

g'(t) = 2t + t^3/2 - (1/2)t^3/2
______________________ multiply thru on top
[1 + t^1/2]²

g'(4) = 2(4) + 4^3/2 - (1/2)(4)^3/2
_________________________ replace t with 4
[1 + 4^1/2]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
_______________________
3y = 4x add

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