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


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


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∂xcan be stated "delf del x"

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

f(x, y, z) = x4 − 3xyz