What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders? Give examples.

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casincal · Accepted Answer

Step 1  Mixed Derivative theorem:&quot; If the function f(x,y) and its partial derivatives fx,fy,fxyandfyx are all defined in any open interval (a,b) and all are continues in the interval, then fxy(a,b)=fyx(a,b)&quot;.  That is, mixed derivative theorem says that the mixed partial derivatives are equal.  Thus, there is no need of calculating all the mixed  partial derivatives. Only one case is enough.  Step 2  For example consider the function f(x,y)=x3y3+x2y+y2x.  Find the first order partial derivatives as follows.  ∂∂x(x3y3+x2y+y2x)=∂∂x(x3y3)+∂∂x(x2y)+∂∂x(y2x)  =3y3x2+2yx+y2  ∂∂y(x3y3+x2y+y2x)=∂∂y(x3y3)+∂∂y(x2y)+∂∂y(y2x)  =3x3y2+x2+2xy  Step 3  Now, find the mixed partial derivatives as,  ∂2∂y∂x(x3y3+x2y+y2x)=∂∂x(3x3y2+x2+2xy)  =9y2x2+2x+2y  ∂2∂x∂y(x3y3+x2y+y2x)=∂∂y(3y3x2+2yx+y2)  =9x2y2+2x+2y  That is, ∂2f∂y∂x=∂2f∂x∂y.

alenahelenash · Accepted Answer

Mathematically, the Mixed Derivative Theorem can be expressed as follows:∂2f∂x∂y=∂2f∂y∂xThis theorem allows us to simplify the process of calculating partial derivatives of second and higher orders by reducing the number of derivatives we need to evaluate.Let&#039;s consider an example to illustrate this concept. Suppose we have a function f(x,y)=x3y2+2xy+3. We want to find the mixed second-order partial derivatives ∂2f∂x∂y and ∂2f∂y∂x.To calculate ∂2f∂x∂y, we differentiate f with respect to x first, and then differentiate the result with respect to y. Applying the Chain Rule, we obtain:∂2f∂x∂y=∂∂y(∂f∂x)=∂∂y(3x2y2+2y)=6xyTo calculate ∂2f∂y∂x, we differentiate f with respect to y first, and then differentiate the result with respect to x. Again using the Chain Rule, we get:∂2f∂y∂x=∂∂x(∂f∂y)=∂∂x(2x+2xy2)=2+2y2As we can see, the mixed second-order partial derivatives ∂2f∂x∂y and ∂2f∂y∂x are equal to 6xy and 2+2y2 respectively. This result confirms the validity of the Mixed Derivative Theorem.By using the Mixed Derivative Theorem, we can simplify the calculation of mixed partial derivatives by choosing the order of differentiation that is most convenient. This theorem is especially useful when dealing with complex functions involving multiple variables, as it allows us to interchange the order of differentiation and obtain the same result.

user_27qwe · Accepted Answer

The Mixed Derivative Theorem states that for a function f(x,y) with continuous second-order partial derivatives, the order of differentiation with respect to two different variables does not matter. In other words, the mixed partial derivatives are equal:∂2f∂x∂y=∂2f∂y∂xThis theorem is helpful in calculating partial derivatives of second and higher orders because it allows us to simplify the process. Instead of calculating all the mixed partial derivatives separately, we only need to compute one of them and then apply the theorem to obtain the others.For example, consider the function f(x,y)=x3y2+2xy. To find the mixed second-order partial derivatives, we can first calculate ∂2f∂x∂y:∂2f∂x∂y=∂∂x(∂f∂y)=∂∂x(3x2y2+2y)=6xy2Then, by applying the Mixed Derivative Theorem, we obtain ∂2f∂y∂x:∂2f∂y∂x=∂∂y(∂f∂x)=∂∂y(3x2y2+2x)=6xyTherefore, ∂2f∂x∂y=∂2f∂y∂x=6xy2=6xy. The Mixed Derivative Theorem allows us to save time and effort by avoiding redundant calculations.

karton · Accepted Answer

Result:∂2f∂x∂y=6x−4y and ∂2f∂y∂x=6x−4ySolution:∂2f∂x∂y=∂2f∂y∂xThis theorem is useful in calculating partial derivatives of second and higher orders because it allows us to interchange the order of differentiation without affecting the result.To illustrate this, let&#039;s consider an example. Suppose we have the function f(x,y)=3x2y−2xy2. We can find the second-order mixed partial derivatives using the Mixed Derivative Theorem.First, we calculate the first-order partial derivatives:∂f∂x=6xy−2y2∂f∂y=3x2−4xyNext, we differentiate these first-order derivatives with respect to the other variable:∂2f∂x∂y=∂∂y(6xy−2y2)=6x−4y∂2f∂y∂x=∂∂x(3x2−4xy)=6x−4yAs we can see, both mixed second-order partial derivatives are equal: ∂2f∂x∂y=6x−4y and ∂2f∂y∂x=6x−4y.

What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders? Give examples.

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