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(3x^2-1)(x^7+2) (6x)(x^7+2) + (3x^2-1)(7x^6) 6x^8 + 12x + 21x^8 -7x^6 27x^8 - 7x^6 + 12x
(2x)(x^3+1) - (x^2-3)(3x^2) 2x^4 + 2x - (3x^4-9x^2) 2x^4 +2x -3x^4 + 9x^2 (-x^4 + 9x^2 +2x) / (x^3+1)^2
function inside another function or, in math terms, a “composite function”
sqrt(x^3+3) x^3+3 is a function inside of another function sqrt()
derivative f(g) = f’(g(x)) * g’(x)
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^2+1 g = x
f’(g(x)) * g’(x) 2(x^3) * 3x
derivative f(g) = f’(g(x)) * g’(x)
derivative f(g) = f’(g(x)) * g’(x)
g = 3x-2x^2 f = x
3(3x-2x^2)^2 * (3-4x) ^ this is in factored form
if there was a + and some other things ???? she would need to factor it out and stuff bunch of things being multiplied is factored form.
(x^2-1)^(2/3)
f = x^(2/3) g = x
2/3(x^2-1)^(-1/3)*(2x)
should prolly try to get rid of negative exponents
2(2x)/(3(x^2-1)^(1/3))
4x/3(x^2-1)^1/3
derivative f(g) = f’(g(x)) * g’(x)
- (7)/(2t-3)
g(t) = -7(2t-3)
f = -7t^-2 g = 2t-3
14(2t-3)^-3 * 2 2*14 = 28 28/(2t-3)^3
derivative f(g) = f’(g(x)) * g’(x)
(3x^3 + 1)(-4x^2-3)
f’g +fg’
(6x^2)((-4x^2-3)^4) + (3x^3+1)(
f = -4x^2 -3 g = x
4(-4x^2-3)^3*-8x (6x^2)((-4x^2-3)^4) + (3x^3+1)(4(-4x^2-3)^3(-8x)) y = (-4x^2-3)^2(9x^2(-4x^2-3) + (-32x)(3x^3+1)) (-4x^2-3)^3(-36x^4 -27x^2 -96x^4 -32x) (-4x^2-3)^3(-132x^4 -27x^2 -32x)
derivative f(g) = f’(g(x)) * g’(x)
function
derivative rule
f(x) = (3x-2)(sqrt(x))
product rule
(sqrt(x))/(3x-2)
quotient
sqrt(3x-2)
chain rule
(3x+2)(sqrt(x^2-4)
product then chain rule
sqrt(x^2-4) / (3x+2)
quotient then chain rule
sqrt((3x+2)^2-4)
chain then chain
sqrt((3x+2)/(x^2-4)))
sqrt((3x+2)(x^2-4))
chain then quotient
chain then product
chain rule: f’(g(x)) * g’(x)
f = x^-2 g = 2x^2 + 3x
f’ = -2x^-3 g’ = 4x + 3
-2(2x^2+3x)^-3 * (4x + 3)
chain rule: f’(g(x)) * g’(x) f = x^1/2 g = 5x^2 + 8
f’ = 1/2x^(-1/2) g’ = 10x
(1/2)(5x^2+8)^(-1/2) * (10x) (10x)(1/2)(5x^2+8)^(-1/2) (5x)(5x^2+8)^-(1/2)
chain rule: f’(g(x)) * g’(x) f = x^3 g = 3-5x
f’ = 3x^2 g’ = -5
3(3-5x)^2 * -5 -15(3-5x)^3
chain rule: f’(g(x)) * g’(x)
f = t^-4 g = 5t^2 -2t
f’ = -4t^-5 g’ = 10t -2
-4(5t^2-2t)^-5 * (10t-2)
f = x^(3/2) g = 9x^-4 + 4x + 7
f’ = 3/2x^1/2 g’ = -36x^-5 + 4
3/2(9x^-4+4x+7)^1/2 * (-36x^-5+4)
(10x^4 - x - 4)^(5/2) f = x^(5/2) g = 10x^4 - x - 4
f’ = (5/2)x^(3/2) g’ = 40x^3 - 1
f’(g(x)) * g’ (5/2)(10x^4-x-4)^(3/2)(40x^3-1)
(5-4x)^5 f = x^5 g = 5-4x
f’ = 5x^4 g’ = -4
-20(5-4x)^4
f x^-4 g 2x^2+3x
f’ -4x^-5 g’ 4x+3
-4(2x^2+3x)^-5(4x+3)
f t^-4 g 7t^2+4t
f’ -4t^-5 g’ 14t + 4
-4(7t^2+4t)^-5 * (14t+4)
f x^1/4 g 5x^4 + 7
f’ 1/4x^(-3/4) g’ 20x
1/4(5x^4+7)^(-3/4)(20x^3) 5x^3(5x^4+7)^(-3/4)
f = 2t^-1/2 g = t^2+5
f’ = -t^(-3/2) g’ = 2t
-(t^2+5)^(-3/2)(2t) -2t(t^2+5)^-3/2
f = x^3 g = 2x^-4 - 3x + 9
f’ = 3x^2 g’ = -8x^-5 - 3
3(2x^-4-3x+9)^2 * (-8x^-5-3)
f’(x)g(x) + g’(x)f(x)
f = (2x^-4-3x+9)^3 g = 2x^3-3x+3
f’ = 3(2x^-4-3x+9)^2 * (-8x^-5-3) g’ = 6x^2 - 3
(-8x^-5-3)(2x^3-3x+3)(3)(2x^-4-3x+9)^2 + (6x^2-3)(2x^-4-3x+9)
f = x^4 g = 6x^4 + 5
f’ = 4x^3 g’ = 24x
4(6x^4+5)^3 * 24x
f’(x)g(x) + g’(x)f(x)
f = 7x g = (6x^4+5)
f’ = 7 g’ = 4(6x^4+5)^3 * 24x
(7)(6x^4+5)^4 + (24x^3)(7x)(4)(6x^4+5)^3
30(35+4t)^2 - 1700t
30(35+4(5))^2 -1700(5) 30(35+20)^2 -8500 30(55)^2 -8500 30(3025)-8500 90750-8500 82250
f = 30t^2 g = 35 +4t
f’ 60t g’ 4
60(35+4t) * 4
240(35+4t) 240(35+4t) -1700
240(35+4(8)) -1700 240(35+32) -1700 240(67) -1700 16080 - 1700 14380
f x^2 f’ = 2x
g = 8 g’ = 9
2g(x) * g’(x)
2(8) * 9 16 * 9 144
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