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

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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

\frac{d}{d y} (\frac{y}{x}) = \frac{1}{x}

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

Simplify

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

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

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

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\int \frac{x + 5}{( x ^{2} + 5 ) ^{3}} d x

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