SHW 6 & 7 Solutions:  DR Practice 1-3


DR Practice 1

`DR Practice 1 Problem 1`

D(f(x)*g(x)) = D(f(x))*g(x)+D(g(x))*f(x)

D((3*x^2-5*x)*(2*x^2+5)) = D(3*x^2-5*x)*(2*x^2+5)+D(2*x^2+5)*(3*x^2-5*x)

D((3*x^2-5*x)*(2*x^2+5)) = (6*x-5)*(2*x^2+5)+4*x*(3*x^2-5*x)

D((3*x^2-5*x)*(2*x^2+5)) = 24*x^3-30*x^2+30*x-25


`DR Practice 1 Problem 2`

D(f(x)*g(x)) = D(f(x))*g(x)+D(g(x))*f(x)

D((x^3+6*x^2-2*x+1)*(2*x^3+6*x)) = D(x^3+6*x^2-2*x+1)*(2*x^3+6*x)+D(2*x^3+6*x)*(x^3+6*x^2-2*x+1)

D((x^3+6*x^2-2*x+1)*(2*x^3+6*x)) = (3*x^2+12*x-2)*(2*x^3+6*x)+(6*x^2+6)*(x^3+6*x^2-2*x+1)

D((x^3+6*x^2-2*x+1)*(2*x^3+6*x)) = 12*x^5+60*x^4+8*x^3+114*x^2-24*x+6


`DR Practice 1 Problem 3`

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D(f(x)/g(x)) = (D(f(x))*g(x)-D(g(x))*f(x))/g(x)^2

D((-4*x^2+x-1)/(-x^3+4*x)) = (D(-4*x^2+x-1)*(-x^3+4*x)-D(-x^3+4*x)*(-4*x^2+x-1))/(-x^3+4*x)^2

D((-4*x^2+x-1)/(-x^3+4*x)) = ((-8*x+1)*(-x^3+4*x)-(-3*x^2+4)*(-4*x^2+x-1))/(-x^3+4*x)^2

D((-4*x^2+x-1)/(-x^3+4*x)) = (-4*x^4+2*x^3-19*x^2+4)/(-x^3+4*x)^2



`DR Practice 1 Problem 4`

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D(f(x)/g(x)) = (D(f(x))*g(x)-D(g(x))*f(x))/g(x)^2

D((2*x^3+6*x)/(x^2+3*x-5)) = (D(2*x^3+6*x)*(x^2+3*x-5)-D(x^2+3*x-5)*(2*x^3+6*x))/(x^2+3*x-5)^2

D((2*x^3+6*x)/(x^2+3*x-5)) = ((6*x^2+6)*(x^2+3*x-5)-(2*x+3)*(2*x^3+6*x))/(x^2+3*x-5)^2

D((2*x^3+6*x)/(x^2+3*x-5)) = (2*x^4+12*x^3-36*x^2-30)/(x^2+3*x-5)^2



`DR Practice 1 Problem 5`

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D(f(x)/g(x)) = (D(f(x))*g(x)-D(g(x))*f(x))/g(x)^2

D((x^5-4*x^4-x^2+10)/(x-3)) = (D(x^5-4*x^4-x^2+10)*(x-3)-D(x-3)*(x^5-4*x^4-x^2+10))/(x-3)^2

D((x^5-4*x^4-x^2+10)/(x-3)) = ((5*x^4-16*x^3-2*x)*(x-3)-x^5+4*x^4+x^2-10)/(x-3)^2

D((x^5-4*x^4-x^2+10)/(x-3)) = (4*x^5-27*x^4+48*x^3-x^2+6*x-10)/(x-3)^2


DR Practice 2

`DR Practice 2 Problem 1`

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D(f(x)*g(x)) = D(f(x))*g(x)+D(g(x))*f(x)

D((5*x^3-3*x^2)*(-4*x^2+1)) = D(5*x^3-3*x^2)*(-4*x^2+1)+D(-4*x^2+1)*(5*x^3-3*x^2)

D((5*x^3-3*x^2)*(-4*x^2+1)) = (15*x^2-6*x)*(-4*x^2+1)-8*x*(5*x^3-3*x^2)

D((5*x^3-3*x^2)*(-4*x^2+1)) = -100*x^4+48*x^3+15*x^2-6*x


`DR Practice 2 Problem 2`

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D(f(x)*g(x)) = D(f(x))*g(x)+D(g(x))*f(x)

D((-x^3+2*x^2-3*x+5)*(x^3+x)) = D(-x^3+2*x^2-3*x+5)*(x^3+x)+D(x^3+x)*(-x^3+2*x^2-3*x+5)

D((-x^3+2*x^2-3*x+5)*(x^3+x)) = (-3*x^2+4*x-3)*(x^3+x)+(3*x^2+1)*(-x^3+2*x^2-3*x+5)

D((-x^3+2*x^2-3*x+5)*(x^3+x)) = -6*x^5+10*x^4-16*x^3+21*x^2-6*x+5


`DR Practice 2 Problem 3`

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D(f(x)/g(x)) = (D(f(x))*g(x)-D(g(x))*f(x))/g(x)^2

D((5*x^3-3*x^2)/(-6*x^2+1)) = (D(5*x^3-3*x^2)*(-6*x^2+1)-D(-6*x^2+1)*(5*x^3-3*x^2))/(-6*x^2+1)^2

D((5*x^3-3*x^2)/(-6*x^2+1)) = ((15*x^2-6*x)*(-6*x^2+1)+12*x*(5*x^3-3*x^2))/(-6*x^2+1)^2

D((5*x^3-3*x^2)/(-6*x^2+1)) = (-30*x^4+15*x^2-6*x)/(-6*x^2+1)^2



`DR Practice 2 Problem 4`

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D(f(x)/g(x)) = (D(f(x))*g(x)-D(g(x))*f(x))/g(x)^2

D((-x^3+2*x^2-3*x+5)/(x^3+x)) = (D(-x^3+2*x^2-3*x+5)*(x^3+x)-D(x^3+x)*(-x^3+2*x^2-3*x+5))/(x^3+x)^2

D((-x^3+2*x^2-3*x+5)/(x^3+x)) = ((-3*x^2+4*x-3)*(x^3+x)-(3*x^2+1)*(-x^3+2*x^2-3*x+5))/(x^3+x)^2

D((-x^3+2*x^2-3*x+5)/(x^3+x)) = (-2*x^4+4*x^3-13*x^2-5)/(x^3+x)^2



`DR Practice 2 Problem 5`

``

D(f(x)/g(x)) = (D(f(x))*g(x)-D(g(x))*f(x))/g(x)^2

D((x^3+x)/(-x^3+2*x^2-3*x+5)) = (D(x^3+x)*(-x^3+2*x^2-3*x+5)-D(-x^3+2*x^2-3*x+5)*(x^3+x))/(-x^3+2*x^2-3*x+5)^2

D((x^3+x)/(-x^3+2*x^2-3*x+5)) = ((3*x^2+1)*(-x^3+2*x^2-3*x+5)-(-3*x^2+4*x-3)*(x^3+x))/(-x^3+2*x^2-3*x+5)^2

D((x^3+x)/(-x^3+2*x^2-3*x+5)) = (2*x^4-4*x^3+13*x^2+5)/(-x^3+2*x^2-3*x+5)^2


DR Practice 3

`DR Practice 3 Problem 1`

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D(f(x)*g(x)) = D(f(x))*g(x)+D(g(x))*f(x)

D((x^2-3)*(x^2-1)^2) = D(x^2-3)*(x^2-1)^2+D((x^2-1)^2)*(x^2-3)

D((x^2-3)*(x^2-1)^2) = 2*x*(x^2-1)^2+4*(x^2-1)*x*(x^2-3)

D((x^2-3)*(x^2-1)^2) = 2*x*(x-1)*(x+1)*(3*x^2-7)


`DR Practice 3 Problem 2`

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D(f(x)*g(x)) = D(f(x))*g(x)+D(g(x))*f(x)

D((x^2+x+1)^2*(-x^5+2)^2) = D((x^2+x+1)^2)*(-x^5+2)^2+D((-x^5+2)^2)*(x^2+x+1)^2

D((x^2+x+1)^2*(-x^5+2)^2) = 2*(x^2+x+1)*(2*x+1)*(-x^5+2)^2-10*(-x^5+2)*x^4*(x^2+x+1)^2

D((x^2+x+1)^2*(-x^5+2)^2) = 2*(x^2+x+1)*(x^5-2)*(7*x^6+6*x^5+5*x^4-4*x-2)


`DR Practice 3 Problem 3`

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D(f(x)/g(x)) = (D(f(x))*g(x)-D(g(x))*f(x))/g(x)^2

D((x^2-3)/(5*x^3-1)^2) = (D(x^2-3)*(5*x^3-1)^2-D((5*x^3-1)^2)*(x^2-3))/(5*x^3-1)^4

D((x^2-3)/(5*x^3-1)^2) = (2*x*(5*x^3-1)^2-30*(5*x^3-1)*x^2*(x^2-3))/(5*x^3-1)^4

D((x^2-3)/(5*x^3-1)^2) = -2*x*(10*x^3-45*x+1)/(5*x^3-1)^3



`DR Practice 3 Problem 4`

``

D(f(x)/g(x)) = (D(f(x))*g(x)-D(g(x))*f(x))/g(x)^2

D((x^2+x+1)^2/(-x^5+2)^2) = (D((x^2+x+1)^2)*(-x^5+2)^2-D((-x^5+2)^2)*(x^2+x+1)^2)/(-x^5+2)^4

D((x^2+x+1)^2/(-x^5+2)^2) = (2*(x^2+x+1)*(2*x+1)*(-x^5+2)^2+10*(-x^5+2)*x^4*(x^2+x+1)^2)/(-x^5+2)^4

D((x^2+x+1)^2/(-x^5+2)^2) = -2*(x^2+x+1)*(x^5-2)*(3*x^6+4*x^5+5*x^4+4*x+2)/(-x^5+2)^4

D((x^2+x+1)^2/(-x^5+2)^2) = -2*(x^2+x+1)*(3*x^6+4*x^5+5*x^4+4*x+2)/(x^5-2)^3



`DR Practice 3 Problem 5`

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D(f(x)/g(x)) = (D(f(x))*g(x)-D(g(x))*f(x))/g(x)^2

D((x-5)^3/(-3*x^2+x-1)^2) = (D((x-5)^3)*(-3*x^2+x-1)^2-D((-3*x^2+x-1)^2)*(x-5)^3)/(-3*x^2+x-1)^4

D((x-5)^3/(-3*x^2+x-1)^2) = (3*(x-5)^2*(-3*x^2+x-1)^2-2*(-3*x^2+x-1)*(-6*x+1)*(x-5)^3)/(-3*x^2+x-1)^4

D((x-5)^3/(-3*x^2+x-1)^2) = -(3*x^2-x+1)*(3*x^2-59*x+7)*(x-5)^2/(-3*x^2+x-1)^4

D((x-5)^3/(-3*x^2+x-1)^2) = -(3*x^2-59*x+7)*(x-5)^2/(3*x^2-x+1)^3