1.7. Elements of calculus: derivatives, integrals, limits, sums, equations

Function	Action
D[f, x]   or	Symbolic calculation of the partial derivative of f with respect to x
D[f, x, y ... ]    or	Sumbolic calculation of the multiple derivative of f with respect to x, then to y etc.
D[f,{x, n }]    or	n -th derivative of f with respect to x (here n=2)
Dt[ f ]	Symbolic calculation of the total derivative of f
Dt[ f, x ]	Symbolic calculation of the total derivative of f on the total derivative of x
D[ f [x] ,x]	Symbolic calculation of the partial derivative of f[x], defined as a function of x
Integrate[f, x] or	Symbolic calculation of the indefinite integral
Integrate[f,{x, a, b}]	Symbolic calculation of the indefinite integral
Integrate[f,{x, a, b}]  //N	Calculates the numerical value of the given definite integral
N [Integrate[f,{x, a, b}]  , n]	Calculates the numerical value of the given definite integral with an accuracy of n digits
Integrate[f,{x, a, b},{y, c, d }]	Symbolic calculation of the double definite integral, where c and d can be functions of a, b. By analogy for the multiple integrals.
Limit[f, x->x0]	Calculates the limit of  f  as x approaches x0
Limit[f, x->Infinity]  or  Limit[f, x->∞]	Calculates the limit of  f  as x approaches ∞
Sum[ f, {i, imin, imax }] or	Calculates the sum of  f  where the index i variates from imin to imax
Sum[ f, {i, imin, imax , di}]	As the previous function but the index i is incremented by a step di
Sum[f,{i, imin, imax},{j,jmin,jmax }]	Multiple summation
Product[ f, {i, imin, imax }] or	Calculates the product
Product[ f, {i, imin, imax , di}]	As the previous function but the index i is incremented by a step di
Product[f,{i,imin, imax},{j,jmin,jmax]	Calculates the multiple product
Series[expression, {x, x0, n}]	Symbolic power series expansion for f about the point up to the n-th degree with the order of the truncation error
Normal[ Series[expr, {x, x0, n}]	As the previous function without the truncation error.
Solve [f==0, x]	Attempts to solve algebraically equation  f = 0 for x. The result is a list of roots, named x
Solve [left_hand_side==right_hand_side, x]	As the previous function for more common case of equation
x1=x /. %[[1]]	Extracts the first root of equation in the variable x1
x2=x /. %%[[2]]	Extracts the second root of equation in the variable x2
N[Solve [f==0, x] , n]	Numerical calculation of the roots of equation with accuracy of n digits
Solve [ {f==0, g==0}, {x, y}]	Attempts to solve exactly the system of simultaneous equations f==0, g==0 for unknowns quantities x,y. The result is a list of numbers and/or expressions for roots.
Eliminate [{left1==right1, left2==right2, ...},{x, y, ...}]	If possible the elimination of the variables x, y, ... from the system of equations is performed and a possible solution is given.
Reduce[{left1==right1, left2==right2, ...},{x, y, ...}]	Analogously as Eliminate function, but with the logical relations.