What is the partial derivative of f(x)= 1/2 x^2 - 5xy?

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For the function f(x) = (1/2)*x^2 - 5xy as there are two variables x and y, the partial derivative is determined by keeping one of them constant.  You did not specify which variable so the answer will be provided for both cases.

The first partial derivative of f(x) with respect...

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For the function f(x) = (1/2)*x^2 - 5xy as there are two variables x and y, the partial derivative is determined by keeping one of them constant.  You did not specify which variable so the answer will be provided for both cases.

The first partial derivative of f(x) with respect to x is:

d ( (1/2)*x^2 - 5xy ) / dx

=> (1/2) *2x - 5y

=> x - 5y

The first partial derivative of f(x) with respect to y is:

d ( (1/2)*x^2 - 5xy ) / dy

=> 0 - 5x

=> -5x

The partial derivative in terms of x is x - 5y.

The partial derivative in terms of y is -5x.

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