What is (\partial)/(\partial y)(x^2ye^{y/z}) ?

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

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(\partial)/(\partial y)(x^2ye^{y/z})

\frac{\partial}{\partial y} (x^{2} y e^{\frac{y}{z}})

x^{2} (e^{\frac{y}{z}} + \frac{e ^{\frac{y}{z}} y}{z})

Solution steps

\frac{\partial}{\partial y} (x^{2} y e^{\frac{y}{z}})

Treat

x, z

as constants

Take the constant out:

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

= x^{2} \frac{\partial}{\partial y} (y e^{\frac{y}{z}})

Apply the Product Rule:

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

f = y, g = e^{\frac{y}{z}}

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

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

\frac{\partial}{\partial y} (e^{\frac{y}{z}}) = \frac{e ^{\frac{y}{z}}}{z}

= x^{2} (1 \cdot e^{\frac{y}{z}} + \frac{e ^{\frac{y}{z}}}{z} y)

Simplify

x^{2} (1 \cdot e^{\frac{y}{z}} + \frac{e ^{\frac{y}{z}}}{z} y) : x^{2} (e^{\frac{y}{z}} + \frac{e ^{\frac{y}{z}} y}{z})

= x^{2} (e^{\frac{y}{z}} + \frac{e ^{\frac{y}{z}} y}{z})

\frac{\partial}{\partial x} (3 x sin (7 x^{2} y))

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

\int_{0}^{1} \frac{3 ^{3 x} - 1}{3 ^{x}} d x

y = (2 x + 1) + \frac{2}{( 2 x + 1 ) ^{3}}

\int_{- 1}^{1} x (1 - x)^{2} d x

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