Exercise calculus.differentiation.quotient
Description
Use the quotient-rule to find the derivative of a polynomial. Only remove parentheses in the numerator.
Derivations
1.
D(x) (5/(x-1))
⇒ calculus.differentiation.quotient((x-1)*D(x) 5-5*D(x) (x-1))/(x-1)^2
⇒ calculus.differentiation.deriv-of-poly-5*D(x) (x-1)/(x-1)^2
⇒ calculus.differentiation.deriv-of-poly-5/(x-1)^2
2.
D(x) (3/(x+2))
⇒ calculus.differentiation.quotient((x+2)*D(x) 3-3*D(x) (x+2))/(x+2)^2
⇒ calculus.differentiation.deriv-of-poly-3*D(x) (x+2)/(x+2)^2
⇒ calculus.differentiation.deriv-of-poly-3/(x+2)^2
3.
D(x) ((-2)/(x-3))
⇒ calculus.differentiation.quotient((x-3)*D(x) (-2)+2*D(x) (x-3))/(x-3)^2
⇒ calculus.differentiation.deriv-of-poly2*D(x) (x-3)/(x-3)^2
⇒ calculus.differentiation.deriv-of-poly2/(x-3)^2
4.
D(x) ((-3)/(x+4))
⇒ calculus.differentiation.quotient((x+4)*D(x) (-3)+3*D(x) (x+4))/(x+4)^2
⇒ calculus.differentiation.deriv-of-poly3*D(x) (x+4)/(x+4)^2
⇒ calculus.differentiation.deriv-of-poly3/(x+4)^2
5.
D(x) (3/(2*x-1))
⇒ calculus.differentiation.quotient((2*x-1)*D(x) 3-3*D(x) (2*x-1))/(2*x-1)^2
⇒ calculus.differentiation.deriv-of-poly-3*D(x) (2*x-1)/(2*x-1)^2
⇒ calculus.differentiation.deriv-of-poly-6/(2*x-1)^2
6.
D(x) (2/(3*x+4))
⇒ calculus.differentiation.quotient((3*x+4)*D(x) 2-2*D(x) (3*x+4))/(3*x+4)^2
⇒ calculus.differentiation.deriv-of-poly-2*D(x) (3*x+4)/(3*x+4)^2
⇒ calculus.differentiation.deriv-of-poly-6/(3*x+4)^2
7.
D(x) ((-4)/(3*x-1))
⇒ calculus.differentiation.quotient((3*x-1)*D(x) (-4)+4*D(x) (3*x-1))/(3*x-1)^2
⇒ calculus.differentiation.deriv-of-poly4*D(x) (3*x-1)/(3*x-1)^2
⇒ calculus.differentiation.deriv-of-poly12/(3*x-1)^2
8.
D(x) ((-2)/(4*x+3))
⇒ calculus.differentiation.quotient((4*x+3)*D(x) (-2)+2*D(x) (4*x+3))/(4*x+3)^2
⇒ calculus.differentiation.deriv-of-poly2*D(x) (4*x+3)/(4*x+3)^2
⇒ calculus.differentiation.deriv-of-poly8/(4*x+3)^2
9.
D(x) ((x+1)/(x-2))
⇒ calculus.differentiation.quotient((x-2)*D(x) (x+1)-(x+1)*D(x) (x-2))/(x-2)^2
⇒ calculus.differentiation.deriv-of-poly(x-2-(x+1)*D(x) (x-2))/(x-2)^2
⇒ calculus.differentiation.deriv-of-poly(x-2-x-1)/(x-2)^2
⇒ algebra.equations.linear.merge-3/(x-2)^2
10.
D(x) ((x-3)/(x+4))
⇒ calculus.differentiation.quotient((x+4)*D(x) (x-3)-(x-3)*D(x) (x+4))/(x+4)^2
⇒ calculus.differentiation.deriv-of-poly(x+4-(x-3)*D(x) (x+4))/(x+4)^2
⇒ calculus.differentiation.deriv-of-poly(x+4-x+3)/(x+4)^2
⇒ algebra.equations.linear.merge7/(x+4)^2
11.
D(x) ((x+5)/(x-1))
⇒ calculus.differentiation.quotient((x-1)*D(x) (x+5)-(x+5)*D(x) (x-1))/(x-1)^2
⇒ calculus.differentiation.deriv-of-poly(x-1-(x+5)*D(x) (x-1))/(x-1)^2
⇒ calculus.differentiation.deriv-of-poly(x-1-x-5)/(x-1)^2
⇒ algebra.equations.linear.merge-6/(x-1)^2
12.
D(x) ((x-2)/(x+1))
⇒ calculus.differentiation.quotient((x+1)*D(x) (x-2)-(x-2)*D(x) (x+1))/(x+1)^2
⇒ calculus.differentiation.deriv-of-poly(x+1-(x-2)*D(x) (x+1))/(x+1)^2
⇒ calculus.differentiation.deriv-of-poly(x+1-x+2)/(x+1)^2
⇒ algebra.equations.linear.merge3/(x+1)^2
13.
D(x) ((2*x+3)/(4*x-1))
⇒ calculus.differentiation.quotient((4*x-1)*D(x) (2*x+3)-(2*x+3)*D(x) (4*x-1))/(4*x-1)^2
⇒ calculus.differentiation.deriv-of-poly((4*x-1)*2-(2*x+3)*D(x) (4*x-1))/(4*x-1)^2
⇒ calculus.differentiation.deriv-of-poly((4*x-1)*2-(2*x+3)*4)/(4*x-1)^2
⇒ algebra.equations.linear.distr-times(8*x-2-(2*x+3)*4)/(4*x-1)^2
⇒ algebra.equations.linear.distr-times(8*x-2-8*x-12)/(4*x-1)^2
⇒ algebra.equations.linear.merge-14/(4*x-1)^2
14.
D(x) ((3*x-1)/(2*x+1))
⇒ calculus.differentiation.quotient((2*x+1)*D(x) (3*x-1)-(3*x-1)*D(x) (2*x+1))/(2*x+1)^2
⇒ calculus.differentiation.deriv-of-poly((2*x+1)*3-(3*x-1)*D(x) (2*x+1))/(2*x+1)^2
⇒ calculus.differentiation.deriv-of-poly((2*x+1)*3-(3*x-1)*2)/(2*x+1)^2
⇒ algebra.equations.linear.distr-times(6*x+3-(3*x-1)*2)/(2*x+1)^2
⇒ algebra.equations.linear.distr-times(6*x+3-6*x+2)/(2*x+1)^2
⇒ algebra.equations.linear.merge5/(2*x+1)^2
15.
D(x) ((4*x+3)/(3*x-2))
⇒ calculus.differentiation.quotient((3*x-2)*D(x) (4*x+3)-(4*x+3)*D(x) (3*x-2))/(3*x-2)^2
⇒ calculus.differentiation.deriv-of-poly((3*x-2)*4-(4*x+3)*D(x) (3*x-2))/(3*x-2)^2
⇒ calculus.differentiation.deriv-of-poly((3*x-2)*4-(4*x+3)*3)/(3*x-2)^2
⇒ algebra.equations.linear.distr-times(12*x-8-(4*x+3)*3)/(3*x-2)^2
⇒ algebra.equations.linear.distr-times(12*x-8-12*x-9)/(3*x-2)^2
⇒ algebra.equations.linear.merge-17/(3*x-2)^2
16.
D(x) ((5*x-2)/(3*x+4))
⇒ calculus.differentiation.quotient((3*x+4)*D(x) (5*x-2)-(5*x-2)*D(x) (3*x+4))/(3*x+4)^2
⇒ calculus.differentiation.deriv-of-poly((3*x+4)*5-(5*x-2)*D(x) (3*x+4))/(3*x+4)^2
⇒ calculus.differentiation.deriv-of-poly((3*x+4)*5-(5*x-2)*3)/(3*x+4)^2
⇒ algebra.equations.linear.distr-times(15*x+20-(5*x-2)*3)/(3*x+4)^2
⇒ algebra.equations.linear.distr-times(15*x+20-15*x+6)/(3*x+4)^2
⇒ algebra.equations.linear.merge26/(3*x+4)^2
17.
D(x) (3*x^2/(2*x^3+4))
⇒ calculus.differentiation.quotient((2*x^3+4)*D(x) (3*x^2)-3*x^2*D(x) (2*x^3+4))/(2*x^3+4)^2
⇒ calculus.differentiation.deriv-of-poly((2*x^3+4)*6*x-3*x^2*D(x) (2*x^3+4))/(2*x^3+4)^2
⇒ calculus.differentiation.deriv-of-poly((2*x^3+4)*6*x-18*x^4)/(2*x^3+4)^2
⇒ algebra.equations.linear.distr-times(12*x^4+24*x-18*x^4)/(2*x^3+4)^2
⇒ algebra.equations.linear.merge(-6*x^4+24*x)/(2*x^3+4)^2
18.
D(x) (2*x^3/(3*x^2-1))
⇒ calculus.differentiation.quotient((3*x^2-1)*D(x) (2*x^3)-2*x^3*D(x) (3*x^2-1))/(3*x^2-1)^2
⇒ calculus.differentiation.deriv-of-poly((3*x^2-1)*6*x^2-2*x^3*D(x) (3*x^2-1))/(3*x^2-1)^2
⇒ calculus.differentiation.deriv-of-poly((3*x^2-1)*6*x^2-12*x^4)/(3*x^2-1)^2
⇒ algebra.equations.linear.distr-times(18*x^4-6*x^2-12*x^4)/(3*x^2-1)^2
⇒ algebra.equations.linear.merge(6*x^4-6*x^2)/(3*x^2-1)^2
19.
D(x) (x^2/(4*x^3-2))
⇒ calculus.differentiation.quotient((4*x^3-2)*D(x) (x^2)-x^2*D(x) (4*x^3-2))/(4*x^3-2)^2
⇒ calculus.differentiation.deriv-of-poly((4*x^3-2)*2*x-x^2*D(x) (4*x^3-2))/(4*x^3-2)^2
⇒ calculus.differentiation.deriv-of-poly((4*x^3-2)*2*x-12*x^4)/(4*x^3-2)^2
⇒ algebra.equations.linear.distr-times(8*x^4-4*x-12*x^4)/(4*x^3-2)^2
⇒ algebra.equations.linear.merge(-4*x^4-4*x)/(4*x^3-2)^2
20.
D(x) (3*x^3/(5*x^2+7))
⇒ calculus.differentiation.quotient((5*x^2+7)*D(x) (3*x^3)-3*x^3*D(x) (5*x^2+7))/(5*x^2+7)^2
⇒ calculus.differentiation.deriv-of-poly((5*x^2+7)*9*x^2-3*x^3*D(x) (5*x^2+7))/(5*x^2+7)^2
⇒ calculus.differentiation.deriv-of-poly((5*x^2+7)*9*x^2-30*x^4)/(5*x^2+7)^2
⇒ algebra.equations.linear.distr-times(45*x^4+63*x^2-30*x^4)/(5*x^2+7)^2
⇒ algebra.equations.linear.merge(15*x^4+63*x^2)/(5*x^2+7)^2
21.
D(x) ((1-x^3)/(x+4))
⇒ calculus.differentiation.quotient((x+4)*D(x) (1-x^3)-(1-x^3)*D(x) (x+4))/(x+4)^2
⇒ calculus.differentiation.deriv-of-poly(-(x+4)*3*x^2-(1-x^3)*D(x) (x+4))/(x+4)^2
⇒ calculus.differentiation.deriv-of-poly(-(x+4)*3*x^2-1+x^3)/(x+4)^2
⇒ algebra.equations.linear.distr-times(-3*x^3-12*x^2-1+x^3)/(x+4)^2
⇒ algebra.equations.linear.merge(-2*x^3-12*x^2-1)/(x+4)^2
22.
D(x) ((x+3)/(2-x^2))
⇒ calculus.differentiation.quotient((2-x^2)*D(x) (x+3)-(x+3)*D(x) (2-x^2))/(2-x^2)^2
⇒ calculus.differentiation.deriv-of-poly(2-x^2-(x+3)*D(x) (2-x^2))/(2-x^2)^2
⇒ calculus.differentiation.deriv-of-poly(2-x^2+(x+3)*2*x)/(2-x^2)^2
⇒ algebra.equations.linear.distr-times(2-x^2+2*x^2+6*x)/(2-x^2)^2
⇒ algebra.equations.linear.merge(2+x^2+6*x)/(2-x^2)^2
23.
D(x) ((1-2*x^3)/(x+1))
⇒ calculus.differentiation.quotient((x+1)*D(x) (1-2*x^3)-(1-2*x^3)*D(x) (x+1))/(x+1)^2
⇒ calculus.differentiation.deriv-of-poly(-(x+1)*6*x^2-(1-2*x^3)*D(x) (x+1))/(x+1)^2
⇒ calculus.differentiation.deriv-of-poly(-(x+1)*6*x^2-1+2*x^3)/(x+1)^2
⇒ algebra.equations.linear.distr-times(-6*x^3-6*x^2-1+2*x^3)/(x+1)^2
⇒ algebra.equations.linear.merge(-4*x^3-6*x^2-1)/(x+1)^2
24.
D(x) ((x+5)/(2-3*x^2))
⇒ calculus.differentiation.quotient((2-3*x^2)*D(x) (x+5)-(x+5)*D(x) (2-3*x^2))/(2-3*x^2)^2
⇒ calculus.differentiation.deriv-of-poly(2-3*x^2-(x+5)*D(x) (2-3*x^2))/(2-3*x^2)^2
⇒ calculus.differentiation.deriv-of-poly(2-3*x^2+(x+5)*6*x)/(2-3*x^2)^2
⇒ algebra.equations.linear.distr-times(2-3*x^2+6*x^2+30*x)/(2-3*x^2)^2
⇒ algebra.equations.linear.merge(2+3*x^2+30*x)/(2-3*x^2)^2
25.
D(x) ((2-x)/(x^2+1)+2*x^3)
⇒ calculus.differentiation.plusD(x) ((2-x)/(x^2+1))+D(x) (2*x^3)
⇒ calculus.differentiation.quotient((x^2+1)*D(x) (2-x)-(2-x)*D(x) (x^2+1))/(x^2+1)^2+D(x) (2*x^3)
⇒ calculus.differentiation.deriv-of-poly(-x^2-1-(2-x)*D(x) (x^2+1))/(x^2+1)^2+D(x) (2*x^3)
⇒ calculus.differentiation.deriv-of-poly(-x^2-1-(2-x)*2*x)/(x^2+1)^2+D(x) (2*x^3)
⇒ calculus.differentiation.deriv-of-poly(-x^2-1-(2-x)*2*x)/(x^2+1)^2+6*x^2
⇒ algebra.equations.linear.distr-times(-x^2-1-4*x+2*x^2)/(x^2+1)^2+6*x^2
⇒ algebra.equations.linear.merge(x^2-1-4*x)/(x^2+1)^2+6*x^2
26.
D(x) ((x^3-3)/(4-x)+x^2)
⇒ calculus.differentiation.plusD(x) ((x^3-3)/(4-x))+D(x) (x^2)
⇒ calculus.differentiation.quotient((4-x)*D(x) (x^3-3)-(x^3-3)*D(x) (4-x))/(4-x)^2+D(x) (x^2)
⇒ calculus.differentiation.deriv-of-poly((4-x)*3*x^2-(x^3-3)*D(x) (4-x))/(4-x)^2+D(x) (x^2)
⇒ calculus.differentiation.deriv-of-poly((4-x)*3*x^2+x^3-3)/(4-x)^2+D(x) (x^2)
⇒ calculus.differentiation.deriv-of-poly((4-x)*3*x^2+x^3-3)/(4-x)^2+2*x
⇒ algebra.equations.linear.distr-times(12*x^2-3*x^3+x^3-3)/(4-x)^2+2*x
⇒ algebra.equations.linear.merge(12*x^2-2*x^3-3)/(4-x)^2+2*x
27.
D(x) ((3-2*x)/(2*x^2-3)+x^3)
⇒ calculus.differentiation.plusD(x) ((3-2*x)/(2*x^2-3))+D(x) (x^3)
⇒ calculus.differentiation.quotient((2*x^2-3)*D(x) (3-2*x)-(3-2*x)*D(x) (2*x^2-3))/(2*x^2-3)^2+D(x) (x^3)
⇒ calculus.differentiation.deriv-of-poly(-(2*x^2-3)*2-(3-2*x)*D(x) (2*x^2-3))/(2*x^2-3)^2+D(x) (x^3)
⇒ calculus.differentiation.deriv-of-poly(-(2*x^2-3)*2-(3-2*x)*4*x)/(2*x^2-3)^2+D(x) (x^3)
⇒ calculus.differentiation.deriv-of-poly(-(2*x^2-3)*2-(3-2*x)*4*x)/(2*x^2-3)^2+3*x^2
⇒ algebra.equations.linear.distr-times(-4*x^2+6-(3-2*x)*4*x)/(2*x^2-3)^2+3*x^2
⇒ algebra.equations.linear.distr-times(-4*x^2+6-12*x+8*x^2)/(2*x^2-3)^2+3*x^2
⇒ algebra.equations.linear.merge(4*x^2+6-12*x)/(2*x^2-3)^2+3*x^2
28.
D(x) ((2*x^3-4)/(6-5*x)+4*x^2)
⇒ calculus.differentiation.plusD(x) ((2*x^3-4)/(6-5*x))+D(x) (4*x^2)
⇒ calculus.differentiation.quotient((6-5*x)*D(x) (2*x^3-4)-(2*x^3-4)*D(x) (6-5*x))/(6-5*x)^2+D(x) (4*x^2)
⇒ calculus.differentiation.deriv-of-poly((6-5*x)*6*x^2-(2*x^3-4)*D(x) (6-5*x))/(6-5*x)^2+D(x) (4*x^2)
⇒ calculus.differentiation.deriv-of-poly((6-5*x)*6*x^2+(2*x^3-4)*5)/(6-5*x)^2+D(x) (4*x^2)
⇒ calculus.differentiation.deriv-of-poly((6-5*x)*6*x^2+(2*x^3-4)*5)/(6-5*x)^2+8*x
⇒ algebra.equations.linear.distr-times(36*x^2-30*x^3+(2*x^3-4)*5)/(6-5*x)^2+8*x
⇒ algebra.equations.linear.distr-times(36*x^2-30*x^3+10*x^3-20)/(6-5*x)^2+8*x
⇒ algebra.equations.linear.merge(36*x^2-20*x^3-20)/(6-5*x)^2+8*x