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

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

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

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

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

Solution steps

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

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

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

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

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

Simplify

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

\int e^{s i n (x)} 7 cos (x) d x

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

\frac{d}{d x} ((x^{2} + y^{2} + z^{2})^{\frac{- 1}{2}})

x \to 0 + lim (x ln (x^{5}))

\frac{d}{d x} (\frac{x + 3}{2 x ^{2} e ^{x}})

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