Consider the following. f(x,y)=\frac{(e^x)}{(1+e^y)} Find the partial derivative (\frac{df}{dy})

Question:

Consider the following. {eq}f(x,y)=\frac{(e^x)}{(1+e^y)}{/eq} Find the partial derivative {eq}(\frac{df}{dy}){/eq}

Answer and Explanation:

Given, {eq}f(x,y)=\frac{(e^x)}{(1+e^y)}{/eq} ,

Differentiating partially with respect to {eq}y {/eq} holding {eq}x {/eq} as a constant,

{eq}\begin{array}{l} \frac{{\partial f}}{{\partial y}} = \frac{\partial }{{\partial y}}\left( {\frac{{{e^x}}}{{1 + {e^y}}}} \right),\\ \frac{{\partial f}}{{\partial y}} = \frac{{\left( {1 + {e^y}} \right) \cdot \frac{\partial }{{\partial y}}\left( {{e^x}} \right) - \frac{\partial }{{\partial y}}\left( {1 + {e^y}} \right) \cdot {e^x}}}{{{{\left( {1 + {e^y}} \right)}^2}}},\\ \frac{{\partial f}}{{\partial y}} = \frac{{\left( {1 + {e^y}} \right) \cdot 0 - {e^y} \cdot {e^x}}}{{{{\left( {1 + {e^y}} \right)}^2}}},\\ \frac{{\partial f}}{{\partial y}} = \frac{{ - {e^{x + y}}}}{{{{\left( {1 + {e^y}} \right)}^2}}}. \end{array} {/eq}


Learn more about this topic:

Partial Derivative: Definition, Rules & Examples

from College Algebra: Help and Review

Chapter 18 / Lesson 12
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