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## 3 Calculus

### 3.1 Differential Operator

Operator: differential expr
Operator: ' expr

The Jacal command `differential` computes the derivative of the expression expr with respect to a generic derivation. It is generic in the sense that nothing is assumed about its effect on the individual variables. The derivation is denoted by a right quote.

```e6 : differential(x^2+y^3);

2
e6: 2 x x' + 3 y  y'

e7 : (x^2+y^3)';

2
e7: 2 x x' + 3 y  y'
```

### 3.2 Derivatives

Command: diff expr var1 …

The Jacal command `diff` computes the derivative of the expression expr with respect to var1, ….

```e6 : diff(x^2+y^3,y);

2
e6: 3 y
```
Command: partial expr var1 …

The Jacal command `partial` computes the partial derivative of the expression expr with respect to var1, ….

```e6 : partial(x^2+@1^3,1);

2
e6: 3 @1
```
Command: PolyDiff poly var1 …

The Jacal command `PolyDiff` computes the derivative of the expression poly with respect to var1, …. It is faster than `diff` but poly must be a polynomial.

### 3.3 Integration

Command: integrate expr var

Returns the indefinite integral of rational expression expr, if that integral is a rational expression containing at most one radical involving var.

```e1 : integrate((3+x^2)*(1+x^2)^(2/3)/(3+6*x^2+3*x^4),x);

2 2/3
x (1 + x )
e1: -------------
2
1 + x

e2 : integrate((1+x^2)^(2/3),x);

;;; could-not-find-algebraic-anti-derivative
non-decreasing-rxd 2 vs 0

e2 : integrate(x*(1+x^2)^(2/3),x);

2        2 2/3
(3 + 3 x ) (1 + x )
e2: ----------------------
10
```
Command: integrate expr var a b

If the indefinite integral of rational expression expr is a rational expression (optionally including a radical involving var), then `integrate` returns the difference of that integral evaluated at b and a.

```e3 : integrate(x*(1+x^2)^(2/3),x,0,1);

2/3
-3 + 6 2
e3: -----------
10
```

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