How do I take the x-partial derivative of -5e^(-x^2 -y^2)(-2x^2 +2xy+1) fx=
These is what I have tried:
fx= [-5e^(-x^2 -y^2)(-4x +2y)] +[(-2x^2 +2xy+1)-5e^(-x^2 -y^2)*-2x]
-fx= [-5e^(-x^2 -y^2)(-4x +2y)] +[(-2x^2 +2xy+1)10xe^(-x^2 -y^2)]
-fx= [-5(-4x +2y)e^(-x^2 -y^2)] +[10x(-2x^2 +2xy+1)e^(-x^2 -y^2)]
-fx= -5(-4x +2y)e^(-x^2 -y^2) +10x(-2x^2 +2xy+1)e^(-x^2 -y^2) Stuck
Treat y as a constant
Take the constant - 5 out:
- 5 d / dx [ e^ ( - x^2 - y^2 ) ( - 2 x^2 + 2 x y +1) ]
Apply the Product Rule:
( f · g )' = f ' · g + f · g'
where
f = e^ ( - x^2 - y^2 ) , g = - 2 x^2 + 2 x y +1
f ' = - 2 e^ ( - x^2 - y^2 ) · x , g' = - 4 x + 2 y
- 5 d / dx [ e^ ( - x^2 - y^2 ) ( - 2 x^2 + 2 x y +1) ] =
- 5 [ - 2 e^ ( - x^2 - y^2 ) · x · ( - 2 x^2 + 2 x y +1 ) + e^ ( - x^2 - y^2 ) · ( - 4 x + 2 y ) ] =
- 5 e^ ( - x^2 - y^2 ) · [ - 2 · x · ( - 2 x^2 + 2 x y +1 ) - 4 x + 2 y ] =
- 5 e^ ( - x^2 - y^2 ) · ( 4 x^3 - 4 x^2 y - 2 x - 4 x + 2 y ) =
- 5 e^ ( - x^2 - y^2 ) · ( 4 x^3 - 4 x^2 y - 6 x + 2 y )
and you can factor out a final factor of 2, if you wish:
-10 e^(-x^2 - y^2) (2x^3 - 2x^2 y - 3x + y)
uhh, There is a typo with the function. It suppose to be 5e^(-x^2 -y^2)(-2x^2 +2xy+1). Still use Bosian same method to solve and end with oobleck answer right?
well duh - if the only change is to turn -5 to 5, then the final answer just changes sign.
To take the x-partial derivative of the given function, you can use the chain rule. Here's how you can proceed:
1. Start with the given function f(x, y) = -5e^(-x^2 - y^2)(-2x^2 + 2xy + 1).
2. Apply the product rule to the first term:
(u * v)' = u' * v + u * v',
where u = -5e^(-x^2 - y^2) and v = (-2x^2 + 2xy + 1).
Calculate the partial derivative of u with respect to x:
du/dx = (-5) * (e^(-x^2 - y^2)) * (-2x),
which simplifies to 10x * e^(-x^2 - y^2).
Keep the second term as it is since it doesn't involve differentiation, i.e., v = (-2x^2 + 2xy + 1).
3. Repeat the same process for the second term:
(u * v)' = u' * v + u * v',
where u = -2x^2 + 2xy + 1 and v = e^(-x^2 - y^2).
Calculate the partial derivative of u with respect to x:
du/dx = -2(2x) + 2y,
which simplifies to -4x + 2y.
Keep the second term as it is since it doesn't involve differentiation, i.e., v = e^(-x^2 - y^2).
4. Combine the derivatives obtained in step 2 and step 3:
fx = du/dx * v + u * dv/dx
= (10x * e^(-x^2 - y^2)) * (-2x^2 + 2xy + 1) + (-2x^2 + 2xy + 1) * (e^(-x^2 - y^2))
5. Simplify the expression if needed.
It seems that you have already attempted these steps, so it is possible that you made errors in the simplification or there might be some cancellation that can still be done. I recommend rechecking your calculation and simplification to see where you might have gone wrong.