## Using Mathematica » Useful Features

## Calculus

Mathematica is useful for many types of mathematical operations, but of particular use are derivatives and integrals.

### Derivatives

Derivatives can be taken using either the function `D`

or the alias `ESC`pd`ESC`.

For example, to take the first derivative of `x^2`

one can use either of the following:

` ``D[x^2,x]`

` ``2 x`

or

` ``Subscript[∂, x]x^2`

` ``2 x`

A derivative can be taken of a derivative:

` ``Subscript[∂, x]Subscript[∂, x]x^2`

` ``2`

But this is equivalent to a simple second derivative which Mathematica can also do:

` ``D[x^2,{x,2}]`

` ``2`

or

` ``Subscript[∂, x, x]x^2`

` ``2`

We can also take derivatives in multiple variables:

` ``Subscript[∂, x, y]1/Sqrt[x^2 + y^2]`

` ``(3 x y)/(x^2 + y^2)^(5/2)`

Here's that but formatted better:

$$\frac{3 x y}{\left(x^2+y^2\right)^{5/2}}$$

And then for fun let's plot this:

` ````
dxy=Subscript[∂, x, y]1/Sqrt[x^2 + y^2];
Plot3D[dxy,{x,-5,5},{y,-5,5}]
```

### Integration

Mathematica also knows how to do more integrals than any chemist is likely to need. The function for integration is `Integrate`

although like with derivatives there is an alias via `ESC`intt`ESC` for indefinite integrals and `ESC`dintt`ESC` for definite integrals

Here's the standard form for `Integrate`

:

` ``Integrate[2,x]`

` ``2 x`

And here's the alias form:

` ``∫2ⅆx`

` ``2 x`

And just to show that Mathematica observes the fundamental theorem of calculus like the rest of us:

` ``∫∫dxyⅆxⅆy`

` ``1/Sqrt[x^2 + y^2]`

Again, here's the same but formatted better:

$$\frac{1}{\sqrt{x^2+y^2}}$$

And we'll plot this too:

` ````
reversedxy=∫∫dxy ⅆxⅆy;
Plot3D[reversedxy,{x,-5,5},{y,-5,5}]
```

### Fun examples:

Mathematica is a great calculus tool and it knows many types of interesting functions so we'll have it integrate the following nasty function for us:

$$P-hash-!!!11573959464626687!!!-hash-4(\cos (\theta ))}{\partial \theta }$$

where $P_i(\cos (\theta ))$ is the $i^{\text{th}}$ Legendre polynomial in $\cos (\theta )$

` ``Integrate[LegendreP[3,Cos[θ]]Cot[θ]D[LegendreP[4,Cos[θ]],θ],{θ,0,π}]`

` ``0`

but Mathematica has absolutely no problem evaluating this integral. It can even tell you that switching the derivative and cot terms makes the integral impossible to evaluate.

` ``Integrate[LegendreP[3,Cos[θ]]D[Cot[θ]LegendreP[4, Cos[θ]],θ], {θ,0,π}]`

$$\int_0^{\pi } \frac{1}{2} \left(-3 \cos (\theta )+5 \cos ^3(\theta )\right) \left(-\frac{1}{8} \left(3-30 \cos ^2(\theta )+35 \cos ^4(\theta )\right) \csc ^2(\theta )+\frac{1}{8} \cot (\theta ) \left(60 \cos (\theta ) \sin (\theta )-140 \cos ^3(\theta ) \sin (\theta )\right)\right) \, d\theta$$

And when it sees that it simply returns the integral unevaluated