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

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

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

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

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

Solution steps

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

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

\frac{\partial}{\partial x} (x + y^{2}) = 1

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

= 1 \cdot e^{x - y} + e^{x - y} (x + y^{2})

Simplify

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

\int_{- 6}^{1} - 2 x^{2} - 10 x + 12 d x

\int ((x - 1)^{2} - (x + 1)^{2}) (x^{3} + 1) d x

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

\frac{d}{d x} (x^{2} arcsinh (9 x))

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