sympy

Use SymPy's solveset for algebraic equations, linsolve for linear systems, nonlinsolve for nonlinear systems, and dsolve for differential equations. Define variables first with symbols(), then pass the equation and variable to the appropriate solver.

Use diff(expr, x) for derivatives and integrate(expr, x) for indefinite integrals, or integrate(expr, (x, a, b)) for definite ones. SymPy returns exact symbolic results rather than numerical approximations.

SymPy performs exact symbolic computation with variables and formulas, while NumPy performs fast numerical computation on arrays. Use lambdify to convert a SymPy expression into a NumPy function when you need both symbolic derivation and numerical speed.

Floats like 0.5 introduce approximation errors in symbolic math. Use Rational(1, 2) or S(1)/2 for exact fractions, and call evalf() only when you need a final numerical value.

Repeatedly calling subs() and evalf() is slow because each call runs symbolic machinery. Convert the expression once with lambdify(x, expr, 'numpy') to get a fast function that works directly on NumPy arrays.

Yes. The latex() function converts expressions to LaTeX strings for documents, and sympy.utilities.codegen produces C or Fortran source code. The autowrap function can even compile expressions into callable Python functions.