Question

Which order of differentiation will calculate f_(xy) faster: x first or y first? (a) f(x,y)=x^(2)+5xy+sinx+7e^(x) (b) f(x,y)=xlnxy

Answer 1

For function (a) f(x,y) = x² + 5xy + sinx + 7
`e^x`, calculating f_(xy) is faster when differentiating with respect to y first.

Step-by-step explanation:

In order to find the mixed partial derivative f_(xy), we need to differentiate the given function twice: first with respect to x and then with respect to y. Alternatively, we can differentiate first with respect to y and then with respect to x. The order in which we differentiate can impact the complexity of calculations.

Let's consider function (a):

[ f(x,y) = x² + 5xy + sin(x) + 7
`e^x` ]

If we differentiate with respect to x first and then y, we get:

[ f_{xy} = (5 + cos(x)) ]

On the other hand, if we differentiate with respect to y first and then x, we get:

[ f_{yx} = (5 + cos(x)) ]

In this case, the order of differentiation does not affect the result; f_(xy) and f_(yx) are equal. However, the choice of order can still impact computational efficiency, and sometimes the expressions can be more straightforward when differentiating in a specific order.

Therefore, for this particular function, whether you differentiate with respect to x first or y first, the calculation for f_(xy) will be equally efficient. The final result will be the same, and the choice of order will not affect the outcome in this specific case.