Partial Derivatives of Functions

In summary, the conversation is about finding the first and second order partial derivatives of the volume of a cone and determining the rate of change of its volume with respect to its height and radius at specific values.
  • #1
HBST
1
0
I am having some trouble solving the problem shown below. Can anyone point me in the right direction? or provide the location of a worked example?

The volume V of a cone of height h and base radius r is given by V=1/3 πr^2 h. The rate of change of its volume V due to stress expansions with respect to its height h and its radius r is to be determined. Derive the first order and second order partial derivatives. Determine the rate of change of its volume with respect to its height h and radius r if the original height h is 1.5 m and radius r is 0.5 m
 
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  • #2
Where, exactly, are you having difficulty? You are given the equation [tex]V= \frac{1}{3}\pi r^2h[/tex] and asked to find the first and second partial derivatives of V with respect to r and h. Can you do that?

The last part of the question simply asks you to evaluate the first derivatives at the specified values of r and h.
 

FAQ: Partial Derivatives of Functions

What are partial derivatives?

A partial derivative of a function is the derivative of that function with respect to one of its independent variables, while holding all other independent variables constant. It measures the rate of change of the function in a specific direction.

Why do we use partial derivatives?

Partial derivatives are useful in multivariable calculus and physics, as they allow us to study the behavior of a function while holding certain variables constant. This is especially useful in optimization problems where we want to find the maximum or minimum value of a function.

How do you find partial derivatives?

To find a partial derivative, you differentiate the function with respect to the specific variable and treat all other variables as constants. This can be done using the same rules of differentiation as for single-variable functions.

What is the difference between partial derivatives and total derivatives?

Partial derivatives only take into account the change in the function in one specific direction, while total derivatives take into account the change in the function in all directions. Total derivatives are used when the function is dependent on multiple variables and can be thought of as the "overall" derivative of the function.

Can partial derivatives be negative?

Yes, partial derivatives can be negative. This occurs when the function is decreasing in the direction of the specific variable being differentiated. A negative partial derivative means that the rate of change of the function is decreasing in that direction.

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