SYMBOLIC DIFFERENTIATION AND INTEGRATION

D[ x^n, x]

D[ x^n, {x,5}]

D[ (x+y)^4, x, y]

Dt[ f[x] g[x] / x^2 , x]

Integrate[ x^n, x]

Integrate[ x^2 / Sqrt[x^2 + a^2], x]

Integrate[ E^x/x, x]

Integrate[ 1/(x^2+1), {x,a,b}]

Integrate[ Sin[x]^2, {x,0,1}]

Integrate[ E^(-x^2-y^2), {x,-Infinity,Infinity},
                         {y,-Infinity,Infinity}]

Integrate[ x^3+y^3, {x,0,1}, {y,0,x} ]

Integrate[ x Sin[x^2+y^2], {x,0,Pi},{y,0,Pi} ]

Integrate[ 1/z, {z,-I,1,I} ]

Sum[ (-x)^i/i, {i,1,12} ]

Product[ (x+k), {k,1,8} ]

myproduct = Product[ (1 - 1/(4 k^2)), {k,1,n} ]

N[ myproduct /. n -> 500 ]

N[2/Pi]

Sum[ 1/i^2, {i,1,Infinity} ]

N[%,9]

N[Pi^2/6,9]

mysum = Sum[ x^i y^j, {i,1,n},{j,1,m} ]

mysum /. {m->3, n->3}

1 + 1 == 2

2 == 0

3 < 2

2 != 1 <= 1 && a+a == 2 a

x^2 + x + 1 == 0

myeqn = %

myeqn

Solve[ x^2 - x - 1 == 0, x]

sols = x /. %

sols[[1]]

Solve[ x^4 + 1 == 0, x]

N[%]

Solve[ x^7 + x^3 - 1 == 0, x] // N

Solve[ Tan[ Log[x] ] == y ,x ]

FindRoot[ Cos[x] == Pi Log[x] , {x,1} ]

Solve[ { x^2 + y^2 == 1, x - y == 0 }, {x,y} ]

Eliminate[ {a x + b y == 0, c x + d y == 1}, y ]

Solve[%,x]

Solve[ {x==1,x==a}, x]

Reduce[ {x==1,x==a}, x]

y''[t] + a^2 y[t] == 0

DSolve[%, y[t], t]

y'[t] - a y[t]^n == 0

DSolve[%, y[t], t]

s1 = Series[ x Tan[x], {x,0,6} ]

s2 = Series[ 1 + x Sin[x], {x,0,10} ]

s1 s2

s1/s2

s1 + Sin[x]

Cos[s1]

InverseSeries[s1]

D[s1,x]

Integrate[s2,x]

Series[ (1 + 2 a x^2)^(1/3), {x,0,4}]

3 %^2 / (2+%)

Normal[%]

Series[ Log[1+ x Tan[y]],{x,0,2},{y,0,2}]

expr = Sin[x] / x

Limit[ expr, x -> 0 ]