What is (\partial)/(\partial x)(1(x-y)e^{1y+5x^2}) ?

The answer to (\partial)/(\partial x)(1(x-y)e^{1y+5x^2}) is 10e^{y+5x^2}x(x-y)+e^{y+5x^2}

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(\partial)/(\partial x)(1(x-y)e^{1y+5x^2})

\frac{\partial}{\partial x} (1 (x - y) e^{1 y + 5 x^{2}})

10 e^{y + 5 x^{2}} x (x - y) + e^{y + 5 x^{2}}

Solution steps

\frac{\partial}{\partial x} (1 \cdot (x - y) e^{1 \cdot y + 5 x^{2}})

Simplify

1 \cdot (x - y) e^{1 \cdot y + 5 x^{2}} : e^{y + 5 x^{2}} (x - y)

= \frac{\partial}{\partial x} (e^{y + 5 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 + 5 x^{2}}, g = x - y

= \frac{\partial}{\partial x} (e^{y + 5 x^{2}}) (x - y) + \frac{\partial}{\partial x} (x - y) e^{y + 5 x^{2}}

\frac{\partial}{\partial x} (e^{y + 5 x^{2}}) = e^{y + 5 x^{2}} \cdot 10 x

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

= e^{y + 5 x^{2}} \cdot 10 x (x - y) + 1 \cdot e^{y + 5 x^{2}}

Simplify

= 10 e^{y + 5 x^{2}} x (x - y) + e^{y + 5 x^{2}}

\frac{\partial}{\partial x} (x cos (\frac{x}{y}))

x \to 0 lim (x^{3} - 2 x^{3} - 4 x + 7)

\frac{\partial}{\partial x} (e^{- 3 x^{2} - y^{2}})

\frac{d}{d x} (x^{2} + 4 y^{2} - 4 x y + 2)

f (x) = (2 x + 1)^{5} + 10 x (2 x + 1)^{4}

What is (\partial)/(\partial x)(1(x-y)e^{1y+5x^2}) ?
The answer to (\partial)/(\partial x)(1(x-y)e^{1y+5x^2}) is 10e^{y+5x^2}x(x-y)+e^{y+5x^2}