Calculate partial derivatives and mixed partial derivatives

In summary, the conversation discusses the concept of partial derivatives and how they are calculated for a function of three variables. It is clarified that differentiating along a single variable does not eliminate the other variables and that a partial derivative is still a function of all three variables. The idea of evaluating the derivative at a specific point is also mentioned.
  • #1
dyn
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Hi. I know how to calculate partial derivatives and mixed partial derivatives such as ∂2f/∂x∂y but I've now become confused about something. If I have a function of 3 variables eg. f(x,y,z) and I calculate ∂x then I am differentiating wrt x while holding y and z constant. Does that mean ∂x then becomes a function of x only ie does ∂x f(x,y,z) = φ( x ) ? If it does then ∂y and ∂z will always be zero but I know this is not the case. I'm confused !
 
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  • #2
dyn said:
Hi. I know how to calculate partial derivatives and mixed partial derivatives such as ∂2f/∂x∂y but I've now become confused about something. If I have a function of 3 variables eg. f(x,y,z) and I calculate ∂x then I am differentiating wrt x while holding y and z constant. Does that mean ∂x then becomes a function of x only ie does ∂x f(x,y,z) = φ( x ) ? If it does then ∂y and ∂z will always be zero but I know this is not the case. I'm confused !
No. It's still a function of three variables.
Differentiating along a single variable simply means to consider the change in values along this coordinate.

You might want to play around a little with Wolfram, e.g. http://www.wolframalpha.com/input/?i=f(x,y)+=+xy^3++4+x^2
Imagine the partial derivation ##\partial_x## as a tangent in ##x-##direction. It still varies with ##y## and ##z##.
 
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  • #3
Thanks. So it is incorrect to write ∂x f(x, y ,z) = g ( x ) ?
 
  • #4
dyn said:
Thanks. So it is incorrect to write ∂x f(x, y ,z) = g ( x ) ?
Unless the variables y and z disappear as a result of differentiating wrt x, they are still there. They might disappear, but don't count on it.
 
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  • #5
dyn said:
Thanks. So it is incorrect to write ∂x f(x, y ,z) = g ( x ) ?
Yes.

You could of course evaluate the derivative at some point ##p=(x_0,y_0,z_0)## and get
$$\frac{\partial}{\partial_x}\bigg{|}_p f(x,y,z) = g(x_0,y_0,z_0)$$
Or if you want to examine the ##x-##component, you could consider ##g(x,y_0,z_0)## and get a function in one variable, because you fixed ##y=y_0## and ##z=z_0##. But this is another issue.
 
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Related to Calculate partial derivatives and mixed partial derivatives

1. What is the purpose of calculating partial derivatives and mixed partial derivatives?

Calculating partial derivatives and mixed partial derivatives allows us to analyze how a function changes with respect to different variables. This is useful in many fields of science, such as physics, engineering, and economics, where we need to understand how a system or process will respond to changes in certain parameters.

2. How do you calculate a partial derivative?

To calculate a partial derivative, we take the derivative of a multivariable function with respect to one of its variables while holding all other variables constant. This can be done by treating the other variables as constants and using the same rules of differentiation as with single variable functions.

3. What is the difference between a partial derivative and a total derivative?

A partial derivative measures the rate of change of a function with respect to one of its variables, while holding all other variables constant. On the other hand, a total derivative measures the overall rate of change of a function with respect to all of its variables simultaneously. In other words, a total derivative takes into account the effects of changes in all variables, while a partial derivative only looks at the effect of one variable.

4. How do you calculate a mixed partial derivative?

A mixed partial derivative is the derivative of a partial derivative with respect to a different variable. To calculate it, we first take the partial derivative with respect to one variable, treating all other variables as constants, and then take the partial derivative of that result with respect to a different variable. In notation, it would look like ∂²f/∂x∂y.

5. What are some real-world applications of calculating partial derivatives and mixed partial derivatives?

Partial derivatives and mixed partial derivatives are used in many fields of science and engineering, such as in optimization problems, fluid dynamics, thermodynamics, and economics. For example, in economics, we can use partial derivatives to analyze how a company's profit changes with respect to different inputs, such as labor and materials. In fluid dynamics, we can use mixed partial derivatives to understand how pressure changes in a fluid when both temperature and density are changing.

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