Evaluating Partial Derivative of g w.r.t. Theta at (2*sqrt(2),pi/4)

In summary, to evaluate the partial derivative of g with respect to theta at (r,[theta])=(2*sqrt(2),pi/4), where g(x,y) = 1/(x+y^2), we use the chain rule and substitute the given polar coordinates for x and y. This gives us the equation r^2=x^2+y^2 and tan[theta]=y/x, which we can then use to find the partial derivative.
  • #1
snoggerT
186
0
use chain rule to evaluate partial derivative of g with respect to theta at (r,[theta])=(2*sqrt(2),pi/4), where g(x,y) = 1/(x+y^2), x=rsin[theta] and y=rcos[theta]


r^2=x^2+y^2 and tan[theta]=y/x

The Attempt at a Solution



I understand how to use the chain rule for partial derivatives, but I can't seem to figure out any of the problems that use polar coordinates. please help.
 
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  • #2
use
x=rcos(theta)
y=rsin(theta)
 
  • #3
m_s_a said:
use
x=rcos(theta)
y=rsin(theta)

- so for every x/y in the equation I get, plug in the polar coordinate for them?
 

What is the meaning of "evaluating partial derivative of g w.r.t. theta"?

The phrase "evaluating partial derivative of g w.r.t. theta" refers to finding the rate of change of the function g with respect to the variable theta at a specific point. It involves taking the derivative of the function with respect to theta and plugging in the given values for theta.

How do I calculate the partial derivative of g w.r.t. theta at a given point?

To calculate the partial derivative of g w.r.t. theta at a given point, you first need to find the derivative of the function g with respect to theta. Then, substitute the given values for theta into the derivative. Finally, evaluate the resulting expression to get the numerical value of the partial derivative at that point.

What do the values (2*sqrt(2),pi/4) represent in the context of this problem?

The values (2*sqrt(2),pi/4) represent a specific point in the function g. The first value, 2*sqrt(2), represents the value of theta, while the second value, pi/4, represents the value of the function itself at that point. This point is used to find the partial derivative of g with respect to theta.

Why is it important to evaluate partial derivatives?

Evaluating partial derivatives allows us to understand how a function changes with respect to one specific variable while holding all other variables constant. This is useful in many fields, including physics, economics, and engineering, as it allows us to predict and analyze the behavior of complex systems.

What are the practical applications of evaluating partial derivatives?

Partial derivatives are used in many real-world applications, such as optimization problems, determining slope and rate of change, and calculating marginal effects in economics. They are also important in understanding the behavior of multi-variable functions in fields such as physics and engineering.

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