Question

Whats the partial derivative of `f(x,y) = 2x+3e^y`?

Answer 1

There are two:

`(del(f(x,y)))/(delx) = 2`

`(del(f(x,y)))/(dely) = 3e^y`

Explanation:

There are two partial derivatives, one with respect to x and the other with respect to y.

To compute the partial derivative with respect to x, you treat any term that does not contain a function of x as if it were a constant that becomes 0, when the derivative is compute and you treat all other factors in terms that contain x as if they were constants.

`(del(f(x,y)))/(delx) = (del(2x+3e^y))/(delx)`

You treat `3e^y` as a constant that will become 0 and the derivative of `2x` with respect to x is just 2:

`(del(f(x,y)))/(delx) = 2`

You do the same thing when you compute the partial derivative with respect to y.

`(del(f(x,y)))/(delx) = (del(2x+3e^y))/(delx)`

You treat the `2x` as if it were a constant and the derivative of `3e^y` with respect to y is `3e^y`:

`(del(f(x,y)))/(delx) = 3e^y`

Answer 2

`(delf)/(delx)=2`,`(delf)/(dely)=3e^y`

Explanation:

`f(x,y)=2x+3e^y`
While finding partial derivative w.r.t.`x`, the change in `f(x,y)` is due to change in the variable `x` alone whereas `y` remains constant.
So, `(delf)/(delx)=2(1)=2`
similarly, while finding partial derivative w.r.t.`y`, the change in `f(x,y)` is due to change in the variable `y` alone whereas `x` remains constant.
Hence, `(delf)/(dely)=3e^y`