Solution assignment 05 Partial differentiation

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Assignment 5

Calculate the first, second and mixed partial derivatives of the following function.

f(x,y)=3e^{x^2}y-xe^y

Solution

For this function we need to apply the chain rule and the product rule.

f_x=3\cdot2x\cdot{e^{x^2}}y-e^y=6xye^{x^2}-e^y

f_y=3e^{x^2}-xe^y

f_{xx}=6ye^{x^2}+12x^2ye^{x^2} (Here we have used the product rule)

f_{yy}=-xe^y

f_{xy}=f_{yx}=6xe^{x^2}-e^y

Again we notice that the mixed derivatives are equal.

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