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Chapter 13: Problem 11

Find the first partial derivatives of the following functions. $$f(x, y)=x e^{y}$$

Short Answer

Expert verified
Question: Find the first partial derivatives of the function f(x, y) = x * e^y with respect to x and y. Answer: The first partial derivatives are: $$\frac{\partial f}{\partial x} = e^{y}$$ $$\frac{\partial f}{\partial y} = x e^{y}$$

Step by step solution

01

Identify the given function

We are given the function: $$f(x, y) = x e^{y}$$
02

Find the partial derivative with respect to x

Keep y constant and take the derivative of f(x, y) with respect to x: $$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x e^{y})$$ Y is treated as a constant in this derivative so we just differentiate x: $$\frac{\partial f}{\partial x} = e^{y}$$
03

Find the partial derivative with respect to y

Now, keep x constant and take the derivative of f(x, y) with respect to y: $$\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x e^{y})$$ X is treated as a constant in this derivative so we focus on differentiating e^y, and using the chain rule we have: $$\frac{\partial f}{\partial y} = x \frac{\partial}{\partial y}(e^{y})$$ $$\frac{\partial f}{\partial y} = x e^{y}$$
04

Write the final solution

The first partial derivatives of the function f(x, y) = x * e^y with respect to x and y are: $$\frac{\partial f}{\partial x} = e^{y}$$ $$\frac{\partial f}{\partial y} = x e^{y}$$

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