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(2x^2-5)^3 f = x^3 g = 2x
3(2x^2-5)^2*(4x) 12(2x^2-5)^2
g = 2x -1 f = x
f’(g(x)) * g’(x) 4(2x-1)^3 * 2 8(2x-1)^3
this is already in factored form so we dont need to multiply it out… i dont know how i would know that.
explicit: y=x^2+3 y’ = 2x dy/dx = 2x
implicit: x + y^2 = 3 ^ d/dx[y^2] = 2y * dy/dx d/dx[x]= 1
dy/dx = _____
whenever variables dont match up we are taking derivative of x and anything else we need derivative to be multiplied at the end *dy/dx
d/dx[y^3] = 3y^2 * dy/dx
6y^2 * dy/dx
6x
1 + 3 * dy/dx
y^3 + y^2 - 5y - x^2 = 4 3y^2*(dy/dx) + 2y*(dy/dx) - 5(dy/dx) - 2x = 0 3y^2*(dy/dx) + 2y*(dy/dx) - 5(dy/dx) = 2x (dy/dx)*(3y^2 + 2y - 5) = 2x (dy/dx) = 2x/(3y^2+2y-5)
2x^2 - y^2 = 1 4x - 2y(dy/dx) = 0 -2y(dy/dx) = -4x 2y(dy/dx) = 4x (dy/dx) = 4x/2y 2x/y
2(1)/1 2
g = 2x -1 f = x
f’(g(x)) * g’(x) 4(2x-1)^3 * 2 8(2x-1)^3
this is already in factored form so we dont need to multiply it out… i dont know how i would know that.
f = x^3 g = x+y
3(x+y)^2 * (1+1(dy/dx))
3(x+y)^2 * (1+1(dy/dx)) = 3x^2 + 3y^2(dy/dx)
(2x)y + x^2(1*dy/dx) + (2y(dy/dx))x + y^2(1) = 0 x^2(dy/dx) + 2y(dy/dx)x = -2xy + y^2 (dy/dx)(x^2+2xy) = -2xy - y^2 dy/dx = (-2xy-y^2)/(x^2+2xy)
dy/dx x^2+y^2 = 4 2x + 2y(dy/dx) = 0
dy/dt t is for time x^2 + y^2 = 4
2x(dx/dt) + 2y(dy/dt) = 0
find dy/dt x = 1 dx/dt = 2
1(dy/dt) = 4x(dy/dt) - 0(dy/dt) 1(dy/dt) = 4x(dx/dt)
x = 1 dx/dt = 2 (dy/dt) = 4(1)(dy/dt) dy/dt = 8(
she kept going and i couldnt get it in time
circles
area of circle = pir
dr/dt = 1 ft per second when r = 4ft what is area looking for da/dt
da/dt = pir^2 da/dt = pi(2r(dr/dt)) da/dt = pi(2(4)(1)) da/dt = 8pi ft^2/s
wall
530 ft
2ft/s
494ft
x
y
x^2 + y^2 = 530^2 2x(dx/dt) + 2y(dy/dt) = 0 2x(dx/dt) = -2y(dy/dt) 2x(dx/dt)/-2y = dy/dt 192(2)/-2(494) 384/-988 -.38 i fucked it up its supposed to be -.78 idk what i did wrong but shes about to move on so ima give up
dx/dt = 2ft/s dy/dt = ? when y is 494 ft from ground
x^2 + 494^2 = 530^2 x = 192
l*w = area of rectangle length increasing 8in/s width decreasing 3in/s length 65 in width 25in dl/dt = 8in/s dw/dt = -3in/s looking for da/dt
d/dt(A) = d/dt(lw) 1 * da/dt = (1* dl/dt)w + l(1(dw/dt)) da/dt = (8)(25) + (65)(-3) da/dt = 200 - 195 da/dt = 5 in^2 /s
6x^3-y^4=2 18x^2dx/dt -4y^3dy/dt
18x^2 * dx/dt -4y^3dy/dt = 0 -4y^3dy/dt = -18x^2dx/dt 4y^3dy/dt = 18x^2*dx/dt
dy/dt = 18x^2/4y^3 * dx/dt dy/dt = 9x^2/2y^3 * dx/dt
x^(1/4) + 5y^(1/2) = 2 1/4x^(-3/4) + (1/2)(5)(y^-1/2)(dy/dx) = 0 (5/2)(y^-1/2)(dy/dx) = -1/4x^(-3/4) dy/dx = (-1/4x^(-3/4))/((5/2)y^(-1/2))
6x^3y^3 = 9 6x^3 * y^3 = 9
u = 6x^3 v = y
u’ = 18x^2 v’ 3y^2 * dy/dx
18x^2 * y^3 + 6x^3 * 3y^2 * dy/dx 18x^2 * y^3 + 18 * x^3 * y^2 * dy/dx = 0 18 * x^3 * y^2 * dy/dx = -18x^2y^3 dy/dx = -18x^2y^3/18x^3y^2 dy/dx = -y/x
3x^3 - 7y^3 = 3 9x^2 - 21y^2 * dy/dx = 0 -21y^2dy/dx = -9x^2 21y^2dy/dx = 9x^2 dy/dx = (9x^2)/(21y^2)
3x^3 - y^2 = -85 9x^2 - 2ydy/dx = 0 -2ydy/dx = -9x^2 2y*dy/dx = 9x^2 dy/dx = 9x^2/2y
9(-3)^2/2(-2) 9(9)/-4 -81/4
area of rectangle = w*l
lrate = 6in/s wrate = -4in/s
dl/dt = 6in/s dw/dt = -4in/s
length = 65in width = 45in
looking for da/dt dA/dt = d/dt(l) * d/dt(w) dA/dt = ldw/dt + wdl/dt dA/dt = (65)(-4) + (45)(6) dA/dt = -260 + 270 dA/dt = 10
l*w = area of rectangle length increasing 8in/s width decreasing 3in/s length 65 in width 25in dl/dt = 8in/s dw/dt = -3in/s looking for da/dt
d/dt(A) = d/dt(lw) 1 * da/dt = (1* dl/dt)w + l(1(dw/dt)) da/dt = (8)(25) + (65)(-3) da/dt = 200 - 195 da/dt = 5 in^2 /s
0.2x^2 + 3xy + 7.1y^2 = 6698.7
d/dx 3xy u = 3x v = y
u’ = 3 v’ = dy/dx 3y + 3x * dy/dx
.4x + 3y + 3x * dy/dx + 14.2y * dy/dx = 0 3x*(dy/dx) + 14.2y * (dy/dx) = -.4x - 3y
dy/dx(3x+14.2y) = -.4x-3y dy/dx = (-.4x-3y)/(3x+14.2y)
(-.4(18)-3(27))/(3(18)+14.2*27) (-7.2-81)/(54+383.4) -88.2 / 437.4
2x^4 - 4xy - y^3 = 23
d/dx -4xy
u = -4x v = y
u’ = -4 v’ = dy/dx
-4y -4x * dy/dx
8x^3 -4y -4xdy/dx -3y^2 * dy/dx = 0 -4xdy/dx -3y^2dy/dx = -8x^3 + 4y 4xdy/dx + 3y^2*dy/dx = 8x^3 - 4y dy/dx(4x+3y^2) = 8x^3 -4y dy/dx = (8x^3-4y)/(4x+3y^2)
(8(2)^3-4(1))/(4(2)+3(1)^2) (8(8)-4)/(8+3) (64-4)/11 60/11 5
6x^2-xy-y^3=6
d/dx -xy u = -x v = y u’ = -1 v’ = dy/dx
-y + -x * dy/dx
12x -y -x * dy/dx -3y^2 * dy/dx = 0 -x dy/dx -3y^2dy/dx = -12x+y xdy/dx + 3y^2dy/dx = 12x-y dy/dx(x+3y^2) = 12x-y dy/dx = (12x-y)/(x+3y^2)
cube volume = length^3 side = 40mm volumerate = -80/s cube surface area = 6(length^2)
dV/dt = 3s^2 * ds/dt dV/dt = 3(40)^2 * ds/dt dV/dt = 3(1600) * ds/dt dV/dt = 4800 * ds/dt -80 = 4800 * ds/dt -1/60 = ds/dt
ds/dt = -1/60
SA = 6s^2 dSA/dt = 12s * ds/dt dSA/dt = 12(40) * -1/60 dSA/dt = 480 * -1/60 dSA/dt = -8
l*w = area of rectangle length increasing 8in/s width decreasing 3in/s length 65 in width 25in dl/dt = 8in/s dw/dt = -3in/s looking for da/dt
d/dt(A) = d/dt(lw) 1 * da/dt = (1* dl/dt)w + l(1(dw/dt)) da/dt = (8)(25) + (65)(-3) da/dt = 200 - 195 da/dt = 5 in^2 /s
wall
530 ft
2ft/s
494ft
x
y
x^2 + y^2 = 530^2 2x(dx/dt) + 2y(dy/dt) = 0 2x(dx/dt) = -2y(dy/dt) 2x(dx/dt)/-2y = dy/dt 192(2)/-2(494) 384/-988 -.38 i fucked it up its supposed to be -.78 idk what i did wrong but shes about to move on so ima give up
dx/dt = 2ft/s dy/dt = ? when y is 494 ft from ground
x^2 + 494^2 = 530^2 x = 192
2105
4ft/s
1376
a^2+b^2=c^2 1376^2+b^2=2105^2 1893376+b^2=4431025 b^2=2537649 b = 1593
1593
x
y
dx/dt = 4ft/s dy/dt = ? when y is 1376 feet away from the ground
y^2+x^2=1593^2 d/dt(y^2+x^2) = d/dt(1593^2) 2ydy/dt+2xdx/dt = 0 2ydy/dt=-2xdx/dt dy/dt = (-2xdx/dt)/2y dy/dt = (xdx/dt)/y dy/dt = -x/y * dx/dt -1593/1376 * 4
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