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derivative: dy/dx = _____ y = x^2 + 3 dy/dx = 2x
differential is isolating dy dy = 2x(dx)
dy/dx = 2x dy = 2x(dx)
du/dx = (x^2+5)^3 chain rule (x^2+5)
3(x^2+5)^2(2x)dx 6x(x^2+5)^2dx
dv/dx = x^2 / x^2 + 9
quotient rule is f’g - fg’ / g
(2x(x^2+9) - x^2(2x) / (x^2+9)^2) * dx ((2x^3 + 18x - 2x^3) / (x^2+9)^2) * dx (18x/(x^2+9)^2)dx
x
x + deltax
deltax dx
deltay
dy
f(x+deltax)
dy/dx = 2x + 5 dy/dx = 2(8) + 5 dy/dx = 18 + 5 dy/dx = 23 dy = 23(dx) dy = 23(-.7) dy = -16.1
deltay = f(x+deltax) - f(x) deltay = 85.79 - 100 deltay = -14.21
deltay - dy -14.21 - ( -16.1 ) -14.21 + 16.1 1.89
i fucked up the differentiation step lol
x
deltay = f(x+deltax) - f(x) deltax = .1 deltay = f(x+.1) - f(x) deltay = f(x+.1) - 81 deltay = 82.81 -81 deltay = 1.81
dy/dx = 2x dy = 2x(dx) dy = 2(9)(.1) dy = 1.8
deltay - dy 1.81 - 1.8 .1
1.81
1.8
.1
f(x+deltax) a= f’(x)dx + f(x) sqrt(16) = 4 f(16-1) a= f’(16)dx + f(16) f(15) a= 1/2(16)^(-1/2)(-1) + sqrt(16) f(15) a= 1/2(16)^(-1/2)(-1) + 4 f(15) a= 1/2(.25)(-1) + 4 f(15) a= -1/8 + 4 f(15) a= 31/8 f(15) a= 3.875
sqrt(15) = 3.873
x = 16 deltax = - 1 f(x) = sqrt(x) f’(x) = 1/2x^(-1/2)
example 5: estimation
using differentials, estimate (2.01)
f(x+deltax) a= f’(x)dx + f(x) f(2+.01) a= f’(2)(.01) + f(2) f(2.01) a= 12(.01) + 8 f(2.01) a= .12 + 8 f(2.01) a= 8.12
f = x^3 f’ = 3x^2
2.01^3 = 8.120601
V = x^3 x = 5 deltax = dx = .01 V’ = 3x^2
f(x+deltax) a= f’(x)dx + f(x) f(5.01) a= f’(5)dx + f(5) f(5.01) a= 3x^2dx
x^2 - 8x - 2
x = -3 deltax = -.7 deltay = y - dy
x^2 - 8x - 2 dy = (2x - 8)dx dy = (2x-8)(-.7) dy = (2(-3)-8)(-.7) dy = (-6-8)(-.7) dy = -14(-.7) dy = 9.8
x^2 - 8x - 2 (-3)^2 - 8(-3) - 2 9 + 24 - 2 31
(-3.7)^2 - 8(-3.7) -2 13.69 + 29.6 -2 41.29
41.29 -31 = 10.29 10.29 - 9.8 = .49
(.4x - 3)dx
(33pir^2)dr
4x^3 + 2x^2 - 10x + 1 (12x^2 + 4x - 10)dx
6pix^2 (12pix)dx
sqrt(9) = 3 x^(1/2)
dy = (1/2x^(-1/2))dx dy = (1/2(9)^(-1/2))(3) dy = (1/2)(1/3)(3) dy = 1/6(3) dy = 3/6 dy = 1/2
3 + 1/2
x^1/2 9^1/2 = 3 12^1/2 = 3.464
chain rule (2x+7)
f = x^2 g = 2x+7
f’(g(x)) * g’(x) 2(2x+7) * 2 4(2x+7) 8x+28
dy = (8x+28)dx dy = (8(-2)+28)(.6) dy = (-16+28)(.6) dy = (12)(.6) dy = 7.2
(2(-2)+7)^2 (-4+7)^2 (3)^2 9
(2(-1.4)+7)^2 (-2.8+7)^2 (4.2)^2 17.64
17.64-9 = 8.64 8.64 - 7.2 = 1.44
is this right? idk
Yes it was! 1.44 is fine!
995 + 4x + .18x
x = 58 dx = 3
.18x^2 + 4x + 995 dy = (.36x + 4)dx dy = (.36(58) + 4)(3) dy = (20.88 + 4)(3) dy = 24.88 * 3 dy = 74.64
995 + 4*(58) + .18*(58)^2 1832.52
995 + 4*(61) + .18*(61)^2 1908.78
1908.78 - 1832.52 = 76.26
76.26 - 74.64 = 1.62
1832.52 + 74.64
102 + 26x + .31x^2 x = 103 dx = 4
.31x^2 + 26x + 102 (.62x + 26)dx (.62(103) + 26)(4) (63.86 + 26)(4) 89.86*4 dy = 359.44
2^3 = 8 so 8
x^(1/3)
1/3x^(-2/3) dx = 4 1/3(8)^(-2/3)(-2) 1/3(1/4)(-2) -1/6
2 - 1/6
chain rule
f = x^2 g = 6x + 9
f’ = 2x g’ = 6
f’(g(x)) * g’(x) 2(6x+9) * 6 12(6x+9) (72x + 108)dx but yknow, t not x.
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