What is the d/(dy)(ye^{y/x}) ?

The d/(dy)(ye^{y/x}) is e^{y/x}+(e^{y/x}y)/x

What is the first d/(dy)(ye^{y/x}) ?

The first d/(dy)(ye^{y/x}) is e^{y/x}+(e^{y/x}y)/x

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\frac{d}{d y} (y e^{\frac{y}{x}})

e^{\frac{y}{x}} + \frac{e ^{\frac{y}{x}} y}{x}

Solution steps

\frac{d}{d y} (y e^{\frac{y}{x}})

Treat

x

as a constant

Apply the Product Rule:

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

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

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

\frac{d y}{d y} = 1

\frac{d}{d y} (e^{\frac{y}{x}}) = \frac{e ^{\frac{y}{x}}}{x}

= 1 \cdot e^{\frac{y}{x}} + \frac{e ^{\frac{y}{x}}}{x} y

Simplify

1 \cdot e^{\frac{y}{x}} + \frac{e ^{\frac{y}{x}}}{x} y : e^{\frac{y}{x}} + \frac{e ^{\frac{y}{x}} y}{x}

= e^{\frac{y}{x}} + \frac{e ^{\frac{y}{x}} y}{x}

\int_{0}^{1} 6 (1 + x)^{6} d x

4 sin (x)

x \to 2 lim (- 3 x + 1)

\frac{\partial}{\partial x} (6 y^{4} x^{5} - 5 y^{3} x^{4})

x \to 0 lim (I n \frac{sin ( x )}{x})

What is the d/(dy)(ye^{y/x}) ?
The d/(dy)(ye^{y/x}) is e^{y/x}+(e^{y/x}y)/x
What is the first d/(dy)(ye^{y/x}) ?
The first d/(dy)(ye^{y/x}) is e^{y/x}+(e^{y/x}y)/x