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

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

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

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

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

Solution steps

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

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^{x y}

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

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

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

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

\frac{d}{d x} (3 + 4 x)

\int \frac{- 30}{( 2 x + 1 ) ^{2}} d x

\int \frac{- sin ( x )}{cos ^{2} ( x )} d x

x \to 1 - lim (x^{2} + 1)

\int \frac{1}{7 - 5 x ^{2}} d x

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