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swat32
3 years ago
11

Differentiating a Logarithmic Function in Exercise, find the derivative of the function. See Examples 1, 2, 3, and 4.

Mathematics
1 answer:
natulia [17]3 years ago
3 0

Answer:

g'(x)=\frac{2e^x-e^{-\frac{x}{2}}}{2(e^x+e^{-\frac{x}{2}})}

Step-by-step explanation:

We are given that a function

g(x)=ln(e^x+e^{-\frac{x}{2}})

We have to find the derivative of function

Differentiate w.r.t x

g'(x)=\frac{1}{e^x+e^{-\frac{x}{2}}}\times (e^x+e^{-\frac{x}{2}}\times (-\frac{1}{2}))

By using formula

\frac{d(lnx)}{dx}=\frac{1}{x}

\frac{d e^x}{dx}=e^x

g'(x)=\frac{e^x-\frac{e^{-\frac{x}{2}}}{2}}{e^x+e^{-\frac{x}{2}}}

Hence, the derivative function

g'(x)=\frac{2e^x-e^{-\frac{x}{2}}}{2(e^x+e^{-\frac{x}{2}})}

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2 years ago
Two different cars each depreciate to 60% of their respective original values. The first car depreciates at an annual rate of 10
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The approximate difference in the ages of the two cars, which  depreciate to 60% of their respective original values, is 1.7 years.

<h3>What is depreciation?</h3>

Depreciation is to decrease in the value of a product in a period of time. This can be given as,

FV=P\left(1-\dfrac{r}{100}\right)^n

Here, (<em>P</em>) is the price of the product, (<em>r</em>) is the rate of annual depreciation and (<em>n</em>) is the number of years.

Two different cars each depreciate to 60% of their respective original values. The first car depreciates at an annual rate of 10%.

Suppose the original price of the first car is x dollars. Thus, the depreciation price of the car is 0.6x. Let the number of year is n_1. Thus, by the above formula for the first car,

0.6x=x\left(1-\dfrac{10}{100}\right)^{n_1}\\0.6=(1-0.1)^{n_1}\\0.6=(0.9)^{n_1}

Take log both the sides as,

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Now, the second car depreciates at an annual rate of 15%. Suppose the original price of the second car is y dollars.

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\log 0.6=\log (0.85)^{n_2}\\\log 0.6={n_2}\log (0.85)\\n_2=\dfrac{\log 0.6}{\log 0.85}\\n_2\approx3.14

The difference in the ages of the two cars is,

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Learn more about the depreciation here;

brainly.com/question/25297296

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