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Find the partial derivate of the given function

– $ z \space = \space  e^xy $

The main objective of this function is to find the partial derivative for the given function.

This question uses the concept of partial derivative. When one of the variables in a function of multiple variables is held constant, its derivative is said to be partial. In differential geometry and vector calculus, partial derivatives are used.

Expert Answer

We have to find the partial derivative of the given function.

Given that:

\[ \space z \space = \space e^xy \]

First, we will find the required partial derivative with respect to $ x $ while we will treat the other term as constant.

So:

\[ \space \frac{ \partial z}{ \partial x} \space = \space \frac{ \partial }{ \partial x} ( e^xy ) \]

\[ \space = \space e^xy \space \frac{ \partial }{ \partial x} (x y) \]

\[ \space = \space e^xy \space (1 \space  . \space y) \]

\[ \space = \space e^xy \space ( y) \]

Thus:

\[ \space = \space ye^xy  \]

Now we have to find the partial derivative with respect to $ y $ while keeping the other term constant, which is $ x $.

So:

\[ \space \frac{ \partial z}{ \partial y} \space = \space \frac{ \partial }{ \partial y } ( e^xy ) \]

\[ \space = \space e^xy \frac{ \partial }{ \partial y } ( x y ) \]

\[ \space = \space e^xy  ( x \space . \space 1 ) \]

\[ \space = \space e^xy  ( x  ) \]

Thus:

\[ \space = \space x e^xy   \]

Numerical Answer

The partial derivative of the given expression with respect to $ x $ is:

\[ \space = \space ye^xy  \]

The partial derivative of the given expression with respect to $ y $ is:

\[ \space = \space x e^xy   \]

Example

Find the partial derivative for the given expression.

\[ \space z \space = \space ( 4 x \space + \space 9)( 8 x \space + \space 5 y ) \]

We have to find the partial derivative for the given function.

Given that:

\[ \space z \space = \space ( 4 x \space + \space 9)( 8 x \space + \space 5 y ) \]

First, we will find the required partial derivative with respect to $ x $ while we will treat the other term as constant.

So using the product rule, we get:

\[ \space \frac{ \partial z}{ \partial x} \space = \space ( 4 )( 8 x \space + \space 5 y ) \space + \space 8(4 x \space + \space 9) \]

\[ \space = \space 32 x \space + \space 20 y \space + \space 32 x \space  + \space  7 2  \]

Thus by simplifying, we get:

\[ \space = \space 6 4 x \space + \space 2 0 y \space + \space 7 2 \]

Now, we will find the required partial derivative with respect to $ y $ while we will treat the other term as constant.

So using the product rule, we get:

\[ \space \frac{ \partial z }{ \partial y }  \space = \space ( 0 )( 8 x \space + \space 5 y ) \space + \space ( 5 )( 4 x \space + \space 9 ) \]

Thus by simplifying, we get:

\[ \space = \space 2 0 x \space + \space 45  \]

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