Chapter 12: Q. 27 (page 944)

Find the first-order partial derivatives for the functions in Exercises 27–36.
$f (x, y) = e^{x} \sin (x y)$

Short Answer

Expert verified

The first order partial derivatives are $f_{x} (x, y) = e^{x} \sin (x y) + y e^{x} \cos (x y) and f_{y} (x, y) = x e^{x} \cos (x y)$

Step by step solution

Given

The given function is: $f (x, y) = e^{x} \sin (x y)$ .

To find

We have to find the first-order partial derivatives.

Calculation

$f (x, y) = e^{x} \sin (x y) f_{x} (x, y) = \frac{\partial}{\partial x} (e^{x} \sin (x y)) f_{x} (x, y) = \frac{\partial}{\partial x} (e^{x}) \sin (x y) + e^{x} \frac{\partial}{\partial x} (\sin (x y)) f_{x} (x, y) = e^{x} \sin (x y) + e^{x} \cos (x y) \frac{\partial}{\partial x} (x y) f_{x} (x, y) = e^{x} \sin (x y) + e^{x} \cos (x y) (y) f_{x} (x, y) = e^{x} \sin (x y) + y e^{x} \cos (x y) f (x, y) = e^{x} \sin (x y) f_{y} (x, y) = \frac{\partial}{\partial y} (e^{x} \sin (x y)) f_{y} (x, y) = \frac{\partial}{\partial y} (e^{x}) \sin (x y) + e^{x} \frac{\partial}{\partial y} (\sin (x y)) f_{y} (x, y) = (0) \sin (x y) + e^{x} (\cos (x y)) \frac{\partial}{\partial y} (x y) f_{y} (x, y) = 0 + e^{x} \cos (x y) (x) f_{y} (x, y) = x e^{x} \cos (x y) Hence, f_{x} (x, y) = e^{x} \sin (x y) + y e^{x} \cos (x y) and f_{y} (x, y) = x e^{x} \cos (x y)$

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Find the first-order partial derivatives for the functions in Exercises 27–36.
$f (x, y) = e^{x} \sin (x y)$

Short Answer

Step by step solution

Given

To find

Calculation

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Find the first-order partial derivatives for the functions in Exercises 27–36.fx,y=exsinxy

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Step by step solution

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Find the first-order partial derivatives for the functions in Exercises 27–36.
$f (x, y) = e^{x} \sin (x y)$