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## Related Questions

###### Asked in Calculus

### What is a partial derivative?

A partial derivative is the derivative of a function of more
than one variable with respect to only one variable. When taking a
partial derivative, the other variables are treated as constants.
For example, the partial derivative of the function f(x,y)=2x2 +
3xy + y2 with respect to x is:
?f/?x = 4x + 3y
here we can see that y terms have been treated as constants when
differentiating.
The partial derivative of f(x,y) with respect to y is:
?f/?y = 3x + 2y
and here, x terms have been treated as constants.

###### Asked in Calculus

### What are spacial derivatives?

###### Asked in Physics, Chemistry, Calculus

### What is the difference between partial derivative and derivative?

Say you have a function of a single variable, f(x). Then there
is no ambiguity about what you are taking the derivative with
respect to (it is always with respect to x).
But what if I have a function of a few variables, f(x,y,z)? Now,
I can take the derivative with respect to x, y, or z. These are
"partial" derivatives, because we are only interested in how the
function varies w.r.t. a single variable, assuming that the other
variables are independent and "frozen".
e.g., Question: how does f vary with respect to y? Answer:
(partial f/partial y)
Now, what if our function again depends on a few variables, but
these variables themselves depend on time: x(t), y(t), z(t) -->
f(x(t),y(t),z(t))? Again, we might ask how f varies w.r.t. one of
the variables x,y,z, in which case we would use partial
derivatives. If we ask how f varies with respect to t, we would do
the following:
df/dt = (partial f/partial x)*dx/dt + (partial f/partial
y)*dy/dt + (partial f/partial z)*dz/dt
df/dt is known as the "total" derivative, which essentially uses
the chain rule to drop the assumption that the other variables are
"frozen" while taking the derivative.
This framework is especially useful in physical problems where I
might want to consider spatial variations of a function (partial
derivatives), as well as the total variation in time (total
derivative).

###### Asked in Calculus, Geometry

### What is geometrical representation of partial derivatives?

The partial derivative of z=f(x,y) have a simple geometrical
representation. Suppose the graph of z = f (x y) is the surface
shown. Consider the partial derivative of f with respect to x at a
point. Holding y constant and varying x, we trace out a curve that
is the intersection of the surface with the vertical plane. The
partial derivative measures the change in z per unit increase in x
along this curve. Thus, it is just the slope of the curve at a
value of x. The geometrical interpretation of is analogous in both
types of derivatives, i.e., Ordinary and Partial
Derivatives

###### Asked in Home Improvement, Airplanes and Aircraft

### What are the Trim conditions in aircraft?

An aircraft is at trim when it is flying under steady-state
conditions (nothing is changing and the airplane is just zipping
along).
More specifically, trim conditions are when Clbeta (partial
derivative of the roll moment coefficient with respect to beta
[sideslip angle]), Cnbeta (partial derivative of the yaw moment
coefficient with respect to beta [sideslip angle]) and Cmbeta
(partial derivative of the pitch moment coefficient with respect to
alpha [angle of attack]) are all equal to zero.

###### Asked in Math and Arithmetic, Calculus, The Difference Between

### What is difference between partial differentiation and total differentiation of the function of two or more variables with example?

total differentiation is closer to implicit differentiation
although you are not solving for dy/dx. in other words:
the total derivative of f(x1,x2,...,xk) with respect to xn=
[df(x1,x2,...,xk)/dx1][dx1/dxn] +
df(x1,x2,...,xk)/dx2[dx2/dxn]+...+df(x1,x2,...,xk)/dxn
+[df(x1,x2,...,xk)/dxn+1][dxn+1/dxn]+...+[df(x1,x2,...,xk)/dxk][dxk/dxn]
however, the partial derivative is not this way.
the partial derivative of f(x1,x2,...,xk) with respect to xn is
just that, can't be expanded.
The chain rule is not the same as total differentiation either.
The chain rule is for partially differentiating f(x1,x2,...,xk)
with respect to a variable not included in the explicit form. In
other words, xn has to be considered a function of this variable
for all integers n. so the total derivative is similar to the chain
rule, but not the same.

###### Asked in Calculus, The Difference Between

### What is the difference between the differentiation of the function and the partial differentiation of the function?

You can differentiate a function when it only contains one
changing variable, like f(x) = x2. It's derivative is f'(x) =
2x.
If a function contains more than one variable, like f(x,y) = x2
+ y2, you can't just "find the derivative" generically because that
doesn't specify what variable to take the derivative with respect
to. Instead, you might "take the derivative with respect to x
(treating y as a constant)" and get fx(x,y) = 2x or "take the
derivative with respect to y (treating x as a constant)" and get
fy(x,y) = 2y.
This is a partial derivative--when you take the derivative of a
function with many variable with respect to one of the variables
while treating the rest as constants.

###### Asked in Math and Arithmetic, Algebra, Calculus, Differential Equations

### Definition of partial differential equation with example?

A partial derivative is the derivative in respect to one
dimension. You can use the rules and tricks of normal
differentiation with partial derivatives if you hold the other
variables as constants, but the actual definition is very similar
to the definition of a normal derivative. In respect to x, it looks
like:
fx(x,y)=[f(x+Δx,y)-f(x,y)]/Δx
and in respect to y:
fy(x,y)=[f(x,y+Δy)-f(x,y)]/Δy
Here's an example. take the function z=3x2+2y
we want to find the partial derivative in respect to x, so we
can use basic differentiation techniques if we treat y as a
constant, so zx'=6x+0 because the derivative of a constant (2y in
this case) is always 0. this applies to any number of dimensions.
if you were finding the partial in respect to a of
f(a,b,c,d,e,f,g), you would just differentiate as normal and hold b
through g as constants.