What is (\partial)/(\partial x)(1(x-y)e^{1y+4x^2}) ?

The answer to (\partial)/(\partial x)(1(x-y)e^{1y+4x^2}) is 8e^{y+4x^2}x(x-y)+e^{y+4x^2}

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(\partial)/(\partial x)(1(x-y)e^{1y+4x^2})

\frac{\partial}{\partial x} (1 (x - y) e^{1 y + 4 x^{2}})

8 e^{y + 4 x^{2}} x (x - y) + e^{y + 4 x^{2}}

Solution steps

\frac{\partial}{\partial x} (1 \cdot (x - y) e^{1 \cdot y + 4 x^{2}})

Simplify

1 \cdot (x - y) e^{1 \cdot y + 4 x^{2}} : e^{y + 4 x^{2}} (x - y)

= \frac{\partial}{\partial x} (e^{y + 4 x^{2}} (x - y))

Treat

y

as a constant

Apply the Product Rule:

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

f = e^{y + 4 x^{2}}, g = x - y

= \frac{\partial}{\partial x} (e^{y + 4 x^{2}}) (x - y) + \frac{\partial}{\partial x} (x - y) e^{y + 4 x^{2}}

\frac{\partial}{\partial x} (e^{y + 4 x^{2}}) = e^{y + 4 x^{2}} \cdot 8 x

\frac{\partial}{\partial x} (x - y) = 1

= e^{y + 4 x^{2}} \cdot 8 x (x - y) + 1 \cdot e^{y + 4 x^{2}}

Simplify

= 8 e^{y + 4 x^{2}} x (x - y) + e^{y + 4 x^{2}}

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What is (\partial)/(\partial x)(1(x-y)e^{1y+4x^2}) ?
The answer to (\partial)/(\partial x)(1(x-y)e^{1y+4x^2}) is 8e^{y+4x^2}x(x-y)+e^{y+4x^2}