What is (\partial)/(\partial x)((x^2+y^2)e^{y^2-x^2}) ?

The answer to (\partial)/(\partial x)((x^2+y^2)e^{y^2-x^2}) is 2e^{y^2-x^2}x-2e^{y^2-x^2}x(x^2+y^2)

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(\partial)/(\partial x)((x^2+y^2)e^{y^2-x^2})

\frac{\partial}{\partial x} ((x^{2} + y^{2}) e^{y^{2} - x^{2}})

2 e^{y^{2} - x^{2}} x - 2 e^{y^{2} - x^{2}} x (x^{2} + y^{2})

Solution steps

\frac{\partial}{\partial x} ((x^{2} + y^{2}) e^{y^{2} - x^{2}})

Treat

y

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{2} + y^{2}, g = e^{y^{2} - x^{2}}

= \frac{\partial}{\partial x} (x^{2} + y^{2}) e^{y^{2} - x^{2}} + \frac{\partial}{\partial x} (e^{y^{2} - x^{2}}) (x^{2} + y^{2})

\frac{\partial}{\partial x} (x^{2} + y^{2}) = 2 x

\frac{\partial}{\partial x} (e^{y^{2} - x^{2}}) = - 2 e^{y^{2} - x^{2}} x

= 2 x e^{y^{2} - x^{2}} + (- 2 e^{y^{2} - x^{2}} x) (x^{2} + y^{2})

Simplify

= 2 e^{y^{2} - x^{2}} x - 2 e^{y^{2} - x^{2}} x (x^{2} + y^{2})

\int 2 - x d x

x \to 3 + lim (4)

\frac{d}{d x} ((3 x^{2} - 2 x)^{2})

x^{2} \frac{d y}{d x} + y = 0

What is (\partial)/(\partial x)((x^2+y^2)e^{y^2-x^2}) ?
The answer to (\partial)/(\partial x)((x^2+y^2)e^{y^2-x^2}) is 2e^{y^2-x^2}x-2e^{y^2-x^2}x(x^2+y^2)