How to find the rate of change of the inputs

Mathematics

1 answer:

iogann1982 [59]3 years ago

4 0

Answer:

To find the average rate of change, we divide the change in the output value by the change in the input value.

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14. Consider the function f(x, y) = yx^4 * e^x². Find fxyxyx (x, y).

Alex787 [66]

The second partial derivatives of $f$ exist everywhere in its domain. By Schwarz's theorem, the mixed second-order partials derivatives are equal:

$f_{xy} = f_{yx}$

Then

$f(x,y) = y x^4 e^{x^2} \\\\ \implies f_y = x^4 e^{x^2} \\\\ \implies f_{yx} = f_{xy} = g(x) \\\\ \implies f_{xyx} = g'(x) \\\\ \implies f_{xyxy} = 0 \\\\ \implies f_{xyxyx} = \boxed{0}$

We don't actually need to compute $g$ and $g'$ to know that they are both free of $y$ . So when we differentiate for a second time with respect to $y$ , the whole thing cancels out.