EV33 said:Homework Statement
I want to show that the partials exist for a certain function.
Homework Equations
My book says that if a function f is differentiable at a point x then the partial derivatives exist.
The Attempt at a Solution
Rather than showing f is differentiable, I am curious if it is sufficient to just use the limit definition of a partial derivative and show that the limit exists?
Thank you for your time.
A partial derivative is a mathematical concept used in multivariable calculus to measure the rate of change of a function with respect to one of its variables, while holding all other variables constant. It essentially tells us how much a function changes in response to a small change in one of its variables.
A regular derivative measures the rate of change of a single-variable function, while a partial derivative measures the rate of change of a multivariable function with respect to one of its variables. This means that a partial derivative takes into account the effects of all other variables on the function, while a regular derivative only considers changes in one variable.
Calculating partial derivatives allows us to better understand the behavior of multivariable functions and their relationships to their input variables. It is also a crucial tool in optimization, as it helps us find the minimum or maximum values of a function.
No, a function cannot be differentiable at a point if it is not continuous at that point. Differentiability requires that the function is continuous and has a well-defined tangent line at that point.
A function is differentiable if all of its partial derivatives exist at every point in its domain. In other words, all of its tangent lines must be well-defined and have a finite slope at every point. This can be determined through the use of the limit definition of a derivative or by checking if the function satisfies the differentiability conditions (the existence of all partial derivatives and their continuity).