Question

Bestimmen Sie die folgende partielle Ableitungen, (a,b und c: Konstante):

\( \mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{ay} e^{y x^{2}}+\mathrm{b} x^{2} e^{y}-\mathrm{c} \ln \left(y x^{2}\right)-5 \mathrm{y} \)

a) \( \frac{\partial f_{(x, y)}}{\partial x}= \)

b) \( \frac{\partial^{2} f_{(x, y)}}{\partial x^{2}}= \)

c) \( \frac{\partial f_{(x, y)}}{\partial y}= \)

d) \( \frac{\partial^{2} f_{(x, y)}}{\partial y^{2}}= \)

e) \( \frac{\partial^{2} f_{(x, y)}}{\partial x \partial y}= \)

Answer 1

f(x) = a·y·e^{y·x^2} + b·x^2·e^y - c·LN(y·x^2) - 5·y

df/dx = 2·a·x·y^2·e^{x^2·y} + 2·b·x·e^y - 2·c/x

d^2f/dx^2 = 4·a·x^2·y^3·e^{x^2·y} + 2·a·y^2·e^{x^2·y} + 2·b·e^y + 2·c/x^2

df/dy = a·x^2·y·e^{x^2·y} + a·e^{x^2·y} + b·x^2·e^y - c/y - 5

d^2f/dy^2 = a·x^4·y·e^{x^2·y} + 2·a·x^2·e^{x^2·y} + b·x^2·e^y + c/y^2

d^2f/dxdy = 2·a·x^3·y^2·e^{x^2·y} + 4·a·x·y·e^{x^2·y} + 2·b·x·e^y