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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

f (x) = x^{3} - 1

\frac{\partial}{\partial x} (4 x^{3} - 5 x^{2} + 8 x)

y^{^{'}} = \frac{2 x + y}{2 x + y + 3}

\int sin^{2} (5 x) d x

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