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Chapter 7: Problem 8

Find \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) for each of the following functions. $$f(x, y)=\frac{e^{x}}{1+e^{y}}$$

Short Answer

Expert verified
The partial derivatives are \(\frac{\partial f}{\partial x} = \frac{e^x}{1 + e^y}\) and \(\frac{\partial f}{\partial y} = \frac{-e^x e^y}{(1 + e^y)^2}\).

Step by step solution

01

Understand the Problem Statement

The function given is \(f(x, y)=\frac{e^{x}}{1+e^{y}}\). We need to find the partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\).
02

Compute \(\frac{\partial f}{\partial x}\)

To find \(\frac{\partial f}{\partial x}\), treat \(y\) as a constant and differentiate with respect to \(x\). Since \(\frac{d}{dx}(e^x) = e^x\) and \(\frac{1}{1 + e^y}\) is constant with respect to \(x\), apply the constant multiple rule to get: \ \( \frac{\frac{e^{x}}{1+e^{y}}}{dx} = \frac{e^x}{1 + e^y} \). Thus, \(\frac{\partial f}{\partial x} = \frac{e^x}{1 + e^y}\).
03

Compute \(\frac{\partial f}{\partial y}\)

To find \(\frac{\partial f}{\partial y}\), treat \(x\) as a constant and differentiate with respect to \(y\). The function can be seen as a quotient: \(u = e^x\) and \(v = 1 + e^y\). The quotient rule states that \(\frac{d}{dy}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dy} - u \frac{dv}{dy}}{v^2} \). Since \(u\) is constant with respect to \(y\), \(\frac{du}{dy} = 0\), and \(\frac{dv}{dy} = e^y\), we have: \ \( \frac{d}{dy}\left(\frac{e^x}{1 + e^y}\right) = \frac{(1 + e^y)(0) - e^x \cdot e^y}{(1 + e^y)^2} = \frac{-e^x e^y}{(1 + e^y)^2} \). Thus, \(\frac{\partial f}{\partial y} = \frac{-e^x e^y}{(1 + e^y)^2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Differentiation
Differentiation is the process of finding the derivative of a function. When dealing with functions of multiple variables, like in our exercise, we often need to find partial derivatives. A partial derivative gives the rate at which the function changes with respect to one variable, keeping all other variables constant. In the given function: \(f(x, y)=\frac{e^{x}}{1+e^{y}}\), we need to find the partial derivatives with respect to both \(x\) and \(y\).
Constant Multiple Rule
The constant multiple rule is a fundamental rule in differentiation. It says if you have a constant multiplied by a function, you can differentiate the function first and then multiply by the constant. For example, if you have \(c \times g(x)\), where \(c\) is a constant, the derivative is \(c \times g'(x)\).

In step 1 of our solution, we treated \(\frac{1}{1+e^y}\) as a constant while differentiating \(e^x\) with respect to \(x\). Since \(\frac{1}{1+e^y}\) does not change with \(x\), the partial derivative \(\frac{\frac{e^{x}}{1+e^{y}}}{dx}\) simply results in \(\frac{e^x}{1 + e^y}\). This is a straightforward application of the constant multiple rule.
Quotient Rule
The quotient rule is used when differentiating a function that is a ratio of two functions. If you have \(u(x)\) over \(v(x)\), the quotient rule states: \[\frac{u}{v}' = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}\] In our exercise, we need the partial derivative of \(f\) with respect to \(y\): \(\frac{\frac{e^{x}}{1+e^{y}}}{dy}\). Here:
  • \(u = e^x\)
  • \(v = 1 + e^y\)
  • \(\frac{du}{dy} = 0\)
  • \(\frac{dv}{dy} = e^y\)
Using the quotient rule, it simplifies to: \[\frac{(1 + e^y)(0) - e^x e^y}{(1 + e^y)^2} = \frac{-e^x e^y}{(1 + e^y)^2}\] This result tells us how the function \(f\) changes as \(y\) changes, while \(x\) is held constant.

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Most popular questions from this chapter

The productivity of a country is given by \(f(x, y)=300 x^{2 / 3} y^{1 / 3},\) where \(x\) and \(y\) are the amount of labor and capital. (a) Compute the marginal productivities of labor and capital when \(x=125\) and \(y=64.\) (b) Use part (a) to determine the approximate effect on the productivity of increasing capital from 64 to 66 units, while keeping labor fixed at 125 units. (c) What would be the approximate effect of decreasing labor from 125 to 124 units while keeping capital fixed at 64 units?

The present value of \(A\) dollars to be paid \(t\) years in the future (assuming a \(5 \%\) continuous interest rate) is \(P(A, t)=A e^{-0.05 t} .\) Find and interpret \(P(100,13.8)\).

The value of residential property for tax purposes is usually much lower than its actual market value. If \(v\) is the market value, the assessed value for real estate taxes might be only \(40 \%\) of \(v .\) Suppose that the property \(\operatorname{tax}, T,\) in a community is given by the function $$T=f(r, v, x)=\frac{r}{100}(.40 v-x)$$ where \(v\) is the market value of a property (in dollars), \(x\) is a homeowner's exemption (a number of dollars depending on the type of property), and \(r\) is the tax rate (stated in dollars per hundred dollars). (a) Determine the real estate tax on a property valued at \(\$ 200,000\) with a homeowner's exemption of \(\$ 5000,\) assuming a tax rate of \(\$ 2.50\) per hundred dollars of net assessed value. (b) Determine the tax duc if the tax rate increases by \(20 \%\) to \(\$ 3.00\) per hundred dollars of net assessed value. Assume the same property value and homeowner's exemption. Does the tax due also increase by \(20 \% ?\)

Find the values of \(x\) and \(y\) that maximize \(x y\) subject to the constraint \(x^{2}-y=3.\)

Let \(R\) be the rectangle consisting of all points \((x, y)\) such that \(0 \leq x \leq 2,2 \leq y \leq 3 .\) Calculate the following double integrals. Interpret each as a volume. $$\iint_{R} e^{y-x} d x d y$$
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