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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

\int_{0}^{1} (x^{2} + x) d x

\int_{- 1}^{2} 3 x 3 x^{2} + 2 d x

\int 10 d x

x \to 2 + lim (x)

\int \frac{21}{1 - cos ( 7 x )} d x

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