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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

\frac{\partial}{\partial x} (x + y^{2}) = 1

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

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

Simplify

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

\frac{6 x d y}{d x} = 2 x e^{x} - 6 y + 6 x^{2}

\frac{d}{d x} (30 ln (5 x + 1))

{\frac{4}{s} + \frac{6}{s ^{5}} - \frac{1}{s + 8}}

\int e^{2 z} d z

\frac{d}{d x} (arctan (x t))

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