Calculus

Derivatives

D[x^3 + Sin[x], x]
D[Sin[x]^2, x]
D[x^n, x]
D[x^2 y^3, x, y]
D[f[x] g[x], x]

3 x^2 + Cos[x]
2 Cos[x] Sin[x]
n x^(-1 + n)
6 x y^2
g[x] f'[x] + f[x] g'[x]

• D[expr, x] differentiates with respect to x; several variables give mixed partial derivatives.
• Undefined functions stay symbolic — the last line is the product rule.

For functions you define, ' works too:

h[x_] := x^3 - 3 x
h'[x]
Solve[h'[x] == 0, x]

-3 + 3 x^2
{{x -> -1}, {x -> 1}}

Integrals

Integrate[x^2, x]
Integrate[x^2, {x, 0, 3}]
Integrate[1/x, x]
Integrate[Exp[-x^2], {x, -Infinity, Infinity}]
NIntegrate[Sin[x^2], {x, 0, 1}]

x^3/3
9
Log[x]
Sqrt[Pi]
0.310268

• Indefinite integrals leave out the constant of integration.
• {x, a, b} gives a definite integral; Infinity works as a limit.
• NIntegrate computes numerically — use it when no formula exists or you only need a number.

Limits, series, and sums

Limit[Sin[x]/x, x -> 0]
Limit[(1 + 1/n)^n, n -> Infinity]
Series[Exp[x], {x, 0, 4}]
Sum[1/n^2, {n, 1, Infinity}]
Sum[k^2, {k, 1, n}]

1
E
1 + x + x^2/2 + x^3/6 + x^4/24 + O[x]^5
Pi^2/6
1/6 n (1 + n) (1 + 2 n)

• Series[f, {x, x0, n}] expands around x0 up to order n; O[x]^5 marks the omitted terms. Normal[series] removes it.
• Sum finds closed forms for many infinite and symbolic sums.

Vector calculus

Grad[x^2 + y^2, {x, y}]
Div[{x, y}, {x, y}]

{2 x, 2 y}
2

Practice

Find the area between y = x and y = x^2 from 0 to 1.

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

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