1) Evaluate the indicated partial derivatives`f(x,y)= x^2ye^(xy)`  where `((del f)/(del x))_(1,1), ((del f)/(del y))_(1,1).`

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the partial derivative `(del f)/(del x), ` hence, you need to differentiate the function with respect to x and you need to consider y as constant, such that:

`(del f)/(del x) =(del(yx^2e^(xy)))/(del x)`

You need to use the product rule such that:

`(del f)/(del x) = y*2x*e^(xy) + yx^2*e^(xy)*y`

`(del f)/(del x) = 2xy*e^(xy) + (xy)^2*e^(xy)`

Factoring out `xy*e^(xy)`  yields:

`(del f)/(del x) = xy*e^(xy)*(2 + xy)`

You need to evaluate `(del f)/(del x) ` at the point `(1,1), ` such that:

`((del f)/(del x))_(1,1) = 1*1*e^(1*1)*(2 + 1*1)`

`((del f)/(del x))_(1,1) = 3e`

You need to find the partial derivative `(del f)/(del y), ` hence, you need to differentiate the function with respect to y and you need to consider x as constant, such that:

`(del f)/(del y) = (del(yx^2e^(xy)))/(del y)`

`(del f)/(del y) = x^2*e^(xy) + x^3*y*e^(xy)`

You need to factor out `x^2*e^(xy)`  such that:

`(del f)/(del y) = x^2*e^(xy)*(1 + xy)`

You need to evaluate `(del f)/(del y)`  at the point `(1,1), ` such that:

`(del f)/(del y)_(1,1) = 1^2*e^(1*1)*(1 + 1*1)`

`((del f)/(del y))_(1,1) = 2e`

Hence, evaluating the partial derivatives at the point `(1,1)`  yields `((del f)/(del x))_(1,1) = 3e ` and `((del f)/(del y))_(1,1) = 2e.`

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