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find f’ f(x) = 6(4e^5x + 1)^3 chain rule
f = 6x^3 g = 4e^5x + 1 f’ = 18x^2 g’ = 20e
f’(g(x)) * g’(x)
18(4e^5x+1)^2 * 20e^5x
ln(x^2) + ln(y) ln(x^2y)
ln(x) + ln(x+5)^(1/2) - ln(x-3)^(1/2) ln(x) + ln(sqrt(x+5)) - ln(sqrt(x-3)) ln(sqrt(x+5)(x)) - ln(sqrt(x-3)) ln(x(sqrt(x+5))/(sqrt(x-3)))
ln(5x) - ln(y) ln(5)+ln(x)-ln(y)
can only do the 2ln(x) + ln(6x) thing if they are being multiplied, not added
x^2+6x+5 (x+5)(x-1) ln((x+5)(x-1)) ln(x+5) + ln(x-1)
ln(2x) = ln(30) 2x = 30 x = 15
e and ln are inverses they undo eachother
x+1 = ln(30) x = ln(30) - 1 8
3x^2 + 1/x
product rule f’g + fg’
f = x^2 g = ln(x) f’ = 2x g’ = 1/x
2x(ln(x)) + x^2(1/x) 2xln(x) + x
ln x 1/x
5, 1/5
ln(5)-1
y = 1/5x + ln(5) - 1
(2x+2)/(x^2+2x+10)
ln(5x+4)^2 + ln(6x-19) 2ln(5x+4) + ln(6x-19) 2(5/5x+4) + 6/(6x-19) 10/(5x+4) + 6/(6x-19)
quotient rule (f’g - fg’) / g^2 f = 2ln(x) g = 18x^3 f’ = 21/x g’ = 54x^2 ((21/x)(18x^3) - 2ln(x)(54x^2))/(18x^3)^2 (36x^2 - 108x^2ln(x))/(324x^6) (36x^2(1-3ln(x)))/(324x^6) (1-3ln(x)) / 9x^4
product rule f’g + fg’ f = 14x^5 g = ln(20x^3-10x^2) f’ = 70x^4 g’ = (60x^2-20x)/(20x^3-10x^2)
70x^4(ln(20x^3-10x^2)) + 14x^5((60x^2-20x)/(20x^3-10x^2))
x^2ln(x) derivative
f’g + fg’
f = x^2 g = ln(x) f’ = 2x g’ = 1/x
2x(ln(x)) + x^2(1/x) 2xln(x) + x x(2lnx + 1) = 0
x(2lnx + 1) = 0
x = 0 0 = 0 not in domain
2lnx + 1 = 0 2lnx = -1 lnx = -1/2 x = e^(-1/2)
f(.1) = -.023 f(e^(-1/2) = -.18 f(2) = 2.77
max (2,2.77) min (e^-1/2, -.18)
ln(19)+2ln(x)+ln(y)
ln((7^2/x)/3) ln(7^2/3x)
ln((x+5)(x+1)) ln(x^2+x+5x+5) ln(x^2+6x+5)
ln(41x^-4y) ln(41)-4ln(x)+ln(y)
ln(8rt(e^2)) ln((e^2)^(1/8)) 1/8ln(e^2) 1/4
7ln(x) + ln(y) ln(yx^7)
ln(x^2-4x-45) - ln(x+5) = ln(9) ln((x^2-4x-45)/(x+5)) = ln(9) (x^2-4x-45)/(x+5) = 9 ((x+5)(x-9))/(x+5) = 9 x-9 = 9 x = 18
1/13(ln(x)+ln(y)) 1/13ln(xy) ln((xy)^(1/13))
2e^(4x-1) = 30 e^(4x-1) = 15 4x-1 = ln(15) 4x = ln(15)+1 x = (ln(15)+1)/4
9x^4ln(x^4) 36x^4ln(x) product rule f = 36x^4 g = ln(x) f’ = 144x^3 g’ = 1/x
f’g + fg’
144x^3(ln(x)) + 36x^4(1/x) 144x^3(ln(x)) + 36x^3 36x^3(4lnx+1)
9x - 3ln(x) 9 - (3/x)
6x^2 - 10x + 8 + ln(x) 12x - 10 + x^-1 12 + x^-2 12 - 1/x^2
6x^2 - 10x + 8 + ln(x) 12x - 10 + x^-1 12 + x^-2 12 - 1/x
12 - 1/x^2 = 0 12 = 1/x^2 12x^2 = 1 x^2 = 1/12 x = sqrt(1/12)
we already know its sqrt(1/12)
6(sqrt(1/12))^2 - 10(sqrt(1/12)) + 8 + ln(sqrt(1/12)) 6(1/12) - 10sqrt(1/12) + 8 + 1/2ln(1/12) 6/12 - 10sqrt(1/12) + 8 + 1/2ln(1/12) 8.5 - 10sqrt(1/12) + 1/2ln(1/12) 5.60284613673 5.603
nah it wanted me to use .289 which wouldve gotten 4.36979740913
quotient rule f’g - fg’ / g
f = 8x g = ln(x) f’ = 8 g’ = 1/x
8ln(x) - 8x(1/x) / (ln(x))^2 8ln(x) - 8 / ln(x)^2 8(ln(x)-1) / ln(x)
8(ln(x)-1) / ln(x)^2 = 0 8(ln(x)-1) = 0 ln(x)-1 = 0 ln(x) = 1 x = e e is like 2.7 so its in range ok so now we solve for 2.1 (8*(2.1))/(ln(2.1)) 22.64
e (8*(e))/(ln(e)) 21.75
4 (8*(4))/(ln(4)) 23.08
min at 3, 21.75 max at 4,23.08
ln(9x^2 - 7x - 10) chain rule f = ln(x) g = 9x^2 - 7x - 10 f’ = 1/x g’ = 18x - 7
f’(g(x)) * g
1/(9x^2-7x-10) * (18x-7) (18x-7)/(9x^2-7x-10)
(18x-7)/(9x^2-7x-10) = 0 18x - 7 = 0 18x = 7 x = 7/18
not in the interval
so evaluate at 11.6 and 13.4 ln((11.6)^2 - 7(11.6) - 10) 3.77
ln((13.4)^2 - 7(13.4) - 10) 4.33
min at 11.6, 3.77 max at 13.4, 4.33
11x^3/ln(x^4) quotient rule f’g - fg’ / g^2 f = 11x^3 g = ln(x^4) f’ = 33x^2 g’ = 4/x (33x^2(ln(x^4)) - 11x^3(4/x)) / ln(x^4)^2 (33x^2(ln(x^4)) - (44x^3/x)) / ln(x^4)^2 (33x^2(ln(x^4)) - 44x^2) / ln(x^4)^2 (33x^2(4)(ln(x) - 44x^2) / (4ln(x))^2 (132x^2(ln(x)) - 44x^2) / (4ln(x))^2 (44x^2(3ln(x) - 1)) / 16(ln(x))^2 (11x^2(3ln(x)-1) / 4ln(x)^2
79x + ln(x^2+5)
chain rule f = ln(x) g = x^2+5 f’ = 1/x g’ = 2x
f’(g(x)) * g’(x)
1/(x^2+5) * 2x 2x/(x^2+5)
79 + 2x/(x^2+5)
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