What is (\partial)/(\partial y)(3(x-y)e^{3y+5x^2}) ?

The answer to (\partial)/(\partial y)(3(x-y)e^{3y+5x^2}) is 3(-e^{3y+5x^2}+3e^{3y+5x^2}(x-y))

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(\partial)/(\partial y)(3(x-y)e^{3y+5x^2})

\frac{\partial}{\partial y} (3 (x - y) e^{3 y + 5 x^{2}})

3 (- e^{3 y + 5 x^{2}} + 3 e^{3 y + 5 x^{2}} (x - y))

Solution steps

\frac{\partial}{\partial y} (3 (x - y) e^{3 y + 5 x^{2}})

Treat

x

as a constant

Take the constant out:

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

= 3 \frac{\partial}{\partial y} ((x - y) e^{3 y + 5 x^{2}})

Apply the Product Rule:

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

f = x - y, g = e^{3 y + 5 x^{2}}

= 3 (\frac{\partial}{\partial y} (x - y) e^{3 y + 5 x^{2}} + \frac{\partial}{\partial y} (e^{3 y + 5 x^{2}}) (x - y))

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

\frac{\partial}{\partial y} (e^{3 y + 5 x^{2}}) = e^{3 y + 5 x^{2}} \cdot 3

= 3 ((- 1) e^{3 y + 5 x^{2}} + e^{3 y + 5 x^{2}} \cdot 3 (x - y))

Simplify

= 3 (- e^{3 y + 5 x^{2}} + 3 e^{3 y + 5 x^{2}} (x - y))

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What is (\partial)/(\partial y)(3(x-y)e^{3y+5x^2}) ?
The answer to (\partial)/(\partial y)(3(x-y)e^{3y+5x^2}) is 3(-e^{3y+5x^2}+3e^{3y+5x^2}(x-y))