# How do you find partial derivatives

## Partial derivative and tangent plane

### Functions with several variables

You already know functions with a variable, for example $ f (x) = x ^ 2 $.

For functions with **several variables** the function depends not only on one variable $ x $, but on several: $ x $, $ y $ and $ z $ or - in general - on $ x_1 $, $ x_2 $, ... We then write a function $ f $ like this:

$ f (x_1; x_2; ...; x_n) $.

In the following we look at examples of functions with two variables. Everything we learn here can also be applied analogously to more than two variables.

Let's start with the function $ f (x; y) = x ^ 2 + y ^ 2 $.

The function graph of this function is a **Area in the room** and is called **Paraboloid** designated. You can think of this as a 3D parabola.

### What is a partial derivative?

If you hold a variable in the function $ f (x; y) = x ^ 2 + y ^ 2 $ with $ y = y_0 $, you get a function $ h (x) = f (x; y_0) = x ^ 2 + y_0 ^ 2 $ in a variable. Its derivative is $ h '(x) = 2x $.

So you can have a **partial derivative** Imagine like this: You derive the function according to one of the two variables and consider the other variable as a constant.

### The first order partial derivatives

The first order partial derivatives are written down as follows:

- $ \ frac {\ partial f} {\ partial x} = f_x $ is the partial first order derivative with respect to $ x $.
- $ \ frac {\ partial f} {\ partial y} = f_y $ is the partial first order derivative with respect to $ y $.

The abbreviations $ f_x $ and $ f_y $ are used more frequently.

The first order partial derivatives can also be written as vectors. This is called **gradient** designated:

$ \ nabla f = \ begin {pmatrix} \ frac {\ partial f} {\ partial x} \ \ frac {\ partial f} {\ partial y} \ end {pmatrix} = \ begin {pmatrix} f_x \ f_y \ end {pmatrix} $.

In the example above, $ f (x) = x ^ 2 + y ^ 2 $ is thus $ f_x = 2x $ and $ f_y = 2y $.

Let's look at another example with the function $ g (x) = 2 \ sin (x) \ times y-3x \ times \ times y ^ 2 $. The partial derivatives are:

- $ g_x = 2 \ cos (x) \ cdot y-3y ^ 2 $ and
- $ g_y = 2 \ sin (x) -6x \ cdot y $.

### The partial derivatives of the second order

As usual, you can also derive functions with several variables several times. The **partial derivatives of the second order** are formed analogously:

- $ \ frac {\ partial ^ 2 f} {\ partial ^ 2 x} = f_ {xx} = 2 $,
- $ \ frac {\ partial ^ 2 f} {\ partial x ~ \ partial y} = f_ {xy} = 0 $,
- $ \ frac {\ partial ^ 2 f} {\ partial y ~ \ partial x} = f_ {yx} = 0 $ and
- $ \ frac {\ partial ^ 2 f} {\ partial ^ 2 y} = f_ {yy} = 2 $.

The partial derivatives of the second order are combined into the so-called **Hessian matrix** (here in abbreviated form):

$ \ text {H} _f = \ begin {pmatrix} f_ {xx} & f_ {xy} \ f_ {yx} & f_ {yy} \ end {pmatrix} $.

For the function $ f (x; y) = x ^ 2 + y ^ 2 $ is the **Hessian matrix** given by this:

$ \ text {H} _f = \ begin {pmatrix} 2 & 0 \ 0 & 2 \ end {pmatrix} $.

### Applications of partial derivative

### Tangent plane at one point

For functions with a variable, you can use the first derivative to set up the equation of a tangent at a point $ x_0 $.

This works in a similar way for functions with several variables. Here you put the equation one **Tangent plane** on.

You need the first order partial derivatives. We look at this again with the example $ f (x; y) = x ^ 2 + y ^ 2 $. First we determine the point of contact. In any case, this must fulfill the functional equation: For $ x_0 = 1 $ and $ y_0 = 1 $ this results in $ z_0 = f (x_0; y_0) = 1 ^ 2 + 1 ^ 2 = 2 $. The point to be examined has the coordinates $ (1 | 1 | 2) $.

A **Tangent plane** is generally described by this equation:

$ z-z_0 = f_x (x_0; y_0) (x-x_0) + f_y (x_0; y_0) (y-y_0) $.

We can plug our point into this equation:

- This leads to the equation $ z-2 = 2 (x-1) +2 (y-1) $.
- You can now transform it as follows: $ z-2 = 2x-2 + 2y-2 $, so $ z-2 = 2x + 2y-4 $.
- You can convert this equation to a plane equation in coordinate form: $ E: ~ 2x + 2y-z = 2 $.

### The necessary condition for extremes

The **necessary condition** For **Extremes** for functions with a variable, $ f '(x) = 0 $. For functions with several variables, the **gradient** be the zero vector. This means that every partial first order derivative must be equal to $ 0 $. For $ f (x) = x ^ 2 + y ^ 2 $ this means:

- $ f_x = 2x = 0 $, so $ x = 0 $.
- $ f_y = 2y = 0 $, so $ y = 0 $.

Incidentally, applies as **sufficient condition** one **Hessian matrix**that their **Determinant** is greater than $ 0 $.

- If $ \ text {H} _ {1; 1} $ is positive, there is a
**local minimum**in front, - if $ \ text {H} _ {1; 1} $ is negative, there is a
**local maximum**before and - otherwise a
**Saddle point**.

The following properties apply to the function $ f (x; y) = x ^ 2 + y ^ 2 $:

- $ \ det (\ text {H} _f) = 4> 0 $ and
- $ \ text {H} _ {1; 1} = 2> 0 $.

So there is a local minimum, as you can see from the area shown.

### The chain rule for functions with several variables

Last but not least, you will learn that **Chain rule for functions with several variables** know. Let $ z = f (x (t); y (t)) $ be a function with several variables. The variables are $ x $ and $ y $, where both $ x = x (t) $ and $ y = y (t) $ each depend on the variable $ t $.

Then $ z $ can be derived from $ t $ as follows:

$ z '(t) = \ frac {\ partial f} {\ partial x} \ cdot x' (t) + \ frac {\ partial f} {\ partial y} \ cdot y '(t) $.

That sure reminds you of them **Chain rule** for functions with a variable.

- Here $ \ frac {\ partial f} {\ partial x} $ and $ \ frac {\ partial f} {\ partial y} $ are the derivatives of the
**external functions**. - $ x '(t) = \ frac {dx} {dt} $ and $ y' (t) = \ frac {dy} {dt} $ are the derivatives of the
**inner functions**.

Again, let's look at an example:

$ z (t) = (x (t)) ^ 2+ (y (t)) ^ 2 = (t ^ 2-2) ^ 2 + (2t + 1) ^ 2 $.

Here $ x (t) = t ^ 2-2 $ and thus the first derivative $ x '(t) = 2t $ as well as $ y (t) = 2t + 1 $ and the first derivative $ y' (t) = $ 2.

Now the chain rule can be applied:

$ \ begin {array} {rcl} z '(t) & = & 2x (t) \ times 2t + 2y (t) \ times 2 \ & = & 2 (t ^ 2-2) \ times 2t + 2 (2t +1) \ cdot 2 \ & = & 4t ^ 3-8t + 8t + 4 \ & = & 4t ^ 3 + 4. \ end {array} $

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