Calc 3 Easy Questions: Partial Derivative of g(x,y)

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In summary, the conversation discusses how to find the partial derivative of a function with respect to y, both using the definition and without it. The use of l'Hospital's rule for limits is suggested and the conversation provides step-by-step instructions on how to solve the problem. Ultimately, the limit is found to be equal to -1.
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don23
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calc 3 easy question!

I am trying to find the partial derivative of the following function with respect to y. I know how to find it without using the definition...but i want to know how to do it both ways. any help??

g(x,y)=x^2*e^-y

I got: lim as h approaches h [(x^2*e^-(y+h))-x^2*e^-y]/h...simplifying it is the hard part. ...any step by step help would be greatly appreciated.
 
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I think you need to use l'Hospital's rule for limits (heard of it?). Anyways, from the point you got to, you can factor x^2*e^(-y) from the expression, and since the only variable with respect to the limit is h, you can take x^2*e^(-y) out of the limit.

Then you're left with lim(e^(-h)-1)/h, which is indeterminate (0/0). Here's where you can use l'Hospital's rule, which states that for an indeterminate ratio of two expressions, the limit of the ratio is equal to the limit of the ratio of the derivatives. Once you do this, the limit is quite friendly, and you can see it is actually equal to -1.

Helpful?
 
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FAQ: Calc 3 Easy Questions: Partial Derivative of g(x,y)

What is a partial derivative?

A partial derivative is a mathematical concept that measures the rate of change of a multivariable function with respect to one of its variables while keeping the other variables constant.

How is a partial derivative calculated?

A partial derivative is calculated by taking the derivative of a multivariable function with respect to one variable while treating all other variables as constants. This can be done using the standard rules of differentiation.

What is the purpose of taking a partial derivative?

The purpose of taking a partial derivative is to understand how a multivariable function changes as one of its variables changes, while holding the others constant. This can help in solving optimization problems and understanding the behavior of complex systems.

Can a function have more than one partial derivative?

Yes, a function can have multiple partial derivatives. This is because there are multiple ways to take the derivative of a multivariable function, depending on which variable is being treated as the independent variable.

How is a partial derivative of g(x,y) different from a regular derivative of a single variable function?

A partial derivative of g(x,y) measures the rate of change of a multivariable function with respect to one of its variables while keeping the others constant. On the other hand, a regular derivative of a single variable function measures the rate of change of a function with respect to its only variable. In other words, a partial derivative takes into account the impact of other variables, while a regular derivative does not.

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