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

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

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

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

x^{^{''}} + 3 x^{^{'}} - 10 x = 6 e^{4 t}

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

\int_{2}^{\infty} \frac{1}{3 x} d x

r^{2} - 13 x^{2}

y^{^{'}} = 3 - 2 x - 0.5 y, y (0) = 1

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