This partial derivative. f(x,y)=(e^x)(y^2). How to solve (∂/∂y),(∂/∂x)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should differentiate the function f with respect to a variable, keeping constant the other variable.

Differentiating f with respect to x, keeping y constant, yields:

`(del f)/(del x) = (del ((e^x)(y^2)))/(del x)`

`(del f)/(del x)=(del(e^x))/(de x)*(y^2) + (e^x)*(del (y^2))/(del x) ` (product rule)

`(del f)/(del x) = (e^x)*(y^2) + 0`

`` `(del f)/(del x) = (e^x)*(y^2)`

Differentiating f with respect to y, keeping x constant, yields:

`(delta f)/(del y) = (del ((e^x)(y^2)))/(del y)`

Using product rule yields:

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

`` `(del f)/(del y) = 2y*(e^x)`

The partial derivatives of the given function are: `(del f)/(del x) = (e^x)*(y^2)`  and `(del f)/(del y) = 2y*(e^x).`