When we discover the slope in the **x** direction (while maintaining **y** fixed) we have found a partial derivative.

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Or we can uncover the steep in the **y** direction (while keeping **x** fixed).

Let"s an initial think about a duty of **one variable** (x):

f(x) = x2

We can discover its derivative making use of the strength Rule:

f’(x) = 2x

But what about a role of **two variables** (x and also y):

f(x, y) = x2 + y3

We can discover its **partial** derivative **with respect to x** as soon as we act **y together a constant** (imagine y is a number prefer 7 or something):

f’x = 2x + 0= 2x

Explanation:

the derivative of x2 (with respect to x) is 2x we**treat y together a constant**, so y3 is likewise a continuous (imagine y=7, then 73=343 is also a constant), and the derivative that a constant is 0

Explanation:

we currently**treat x as a constant**, therefore x2 is likewise a constant, and the derivative the a continuous is 0the derivative the y3 (with respect come y) is 3y2

That is all there is to it. Just remember come treat **all various other variables together if they space constants**.

### Holding A change Constant

So what go "holding a variable constant" watch like?

### Example: the volume of a cylinder is V = π r2 h

We can write the in "multi variable" type as

f(r, h) = π r2 h

For the partial derivative v respect come r we hold **h constant**, and also rchanges:

f’r = π (2r) h = 2πrh

(The derivative the r2 through respect to r is 2r, and also π and h space constants)

It says "as only the radius changes (by the tiniest amount), the volume alters by 2πrh"

It is like we include a skin v a circle"s circumference (2πr) and also a elevation of h.

For the partial derivative v respect come h we host **r constant**:

f’h = π r2 (1)= πr2

(π and also r2 are constants, and the derivative the h with respect come h is 1)

It states "as just the height transforms (by the tiniest amount), the volume alters by πr2"

It is like we add the thinnest decaying on peak with a circle"s area the πr2.

Let"s see another example.

### Example: The surface area the a square prism.

The surface contains the top and also bottom with areas of **x2** each, and 4 political parties of area **xy** each:

f(x, y) = 2x2 + 4xy

f’x = 4x + 4y

f’y = 0 + 4x = 4x

### Three or an ext Variables

We can have 3 or an ext variables. Just discover the partial derivative of each variable subsequently whiletreating** all other variables as constants**.

### Example: The volume the a cube with a square prism reduced out from it.

f(x, y, z) = z3 − x2y

f’x = 0 − 2xy = −2xy

f’y = 0 − x2 = −x2

f’z = 3z2 − 0 = 3z2

When over there are many x"s and also y"s the can gain confusing, therefore a mental trick is to readjust the "constant" variables right into letters like "c" or "k" that look favor constants.

### Example: f(x, y) = y3sin(x) + x2tan(y)

It has x"s and also y"s almost everywhere the place! for this reason let us try the letter change trick.

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With respect come x we can readjust "y" to "k":

f(x, y) = k3sin(x) + x2tan(k)

f’x = k3cos(x) + 2x tan(k)

But remember to turn it earlier again!

f’x = y3cos(x) + 2x tan(y)

Likewise with respect come y we revolve the "x" into a "k":

f(x, y) = y3sin(k) + k2tan(y)

f’y = 3y2sin(k) + k2sec2(y)

f’y = 3y2sin(x) + x2sec2(y)

But only do this if you have trouble remembering, as it is a tiny extra work.

**Notation**: we have actually used **f’x** to median "the partial derivative with respect to x", yet another really common notation is to use a funny backwards d (∂) like this:

*∂f***∂x** = 2x

Which is the exact same as:

f’x = 2x

∂ is dubbed "del" or "dee" or "curly dee"

So *∂f***∂x**can be stated "delf del x"

### Example: uncover the partial derivatives of**f(x, y, z) = x4 − 3xyz** making use of "curly dee" notation

f(x, y, z) = x4 − 3xyz