What is (\partial)/(\partial x)(3(x-y)e^{3y+3x^2}) ?

The answer to (\partial)/(\partial x)(3(x-y)e^{3y+3x^2}) is 3(e^{3y+3x^2}+6e^{3y+3x^2}x(x-y))

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

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

3 (e^{3 y + 3 x^{2}} + 6 e^{3 y + 3 x^{2}} x (x - y))

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = x - y, g = e^{3 y + 3 x^{2}}

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

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

\frac{\partial}{\partial x} (e^{3 y + 3 x^{2}}) = e^{3 y + 3 x^{2}} \cdot 6 x

= 3 (1 \cdot e^{3 y + 3 x^{2}} + e^{3 y + 3 x^{2}} \cdot 6 x (x - y))

Simplify

= 3 (e^{3 y + 3 x^{2}} + 6 e^{3 y + 3 x^{2}} x (x - y))

What is (\partial)/(\partial x)(3(x-y)e^{3y+3x^2}) ?
The answer to (\partial)/(\partial x)(3(x-y)e^{3y+3x^2}) is 3(e^{3y+3x^2}+6e^{3y+3x^2}x(x-y))