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Alik [6]
3 years ago
12

For the following exercises, use logarithmic differentiation to find dy/dx.

Mathematics
1 answer:
kondaur [170]3 years ago
8 0

If y=x^{\log_2(x)}, then taking the logarithm of both sides gives

\ln(y) = \ln\left(x^{\log_2(x)}\right) = \log_2(x) \ln(x)

Differentiate both sides with respect to x :

\dfrac{d\ln(y)}{dx} = \dfrac{d\log_2(x)}{dx}\ln(x) + \log_2(x)\dfrac{d\ln(x)}{dx}

\dfrac1y \dfrac{dy}{dx} = \dfrac{d\log_2(x)}{dx}\ln(x) + \dfrac{\log_2(x)}{x}

Now if z=\log_2(x), then 2^z=x. Rewrite

2^z = e^{\ln(2^z)} = e^{\ln(2)z}

Then by the chain rule,

\dfrac{d2^z}{dx} = \dfrac{dx}{dx}

\dfrac{de^{\ln(2)z}}{dx} = 1

e^{\ln(2)z} \ln(2) \dfrac{dz}{dx}= 1

\dfrac{dz}{dx} = \dfrac{1}{e^{\ln(2)z}\ln(2)}

\dfrac{dz}{dx} = \dfrac{1}{2^z \ln(2)}

\dfrac{d\log_2(x)}{dx} = \dfrac{1}{\ln(2)x}

So we have

\dfrac1y \dfrac{dy}{dx} = \dfrac{\ln(x)}{\ln(2)x}+ \dfrac{\log_2(x)}{x}

\dfrac1y \dfrac{dy}{dx} = \dfrac{\log_2(x)}{x}+ \dfrac{\log_2(x)}{x}

\dfrac1y \dfrac{dy}{dx} = \dfrac{2\log_2(x)}{x}

\dfrac1y \dfrac{dy}{dx} = \dfrac{\log_2(x^2)}{x}

\dfrac{dy}{dx} = \dfrac{y\log_2(x^2)}{x}

Replace y :

\dfrac{dy}{dx} = \dfrac{x^{\log_2(x)}\log_2(x^2)}{x}

\dfrac{dy}{dx} = x^{\log_2(x)-1}\log_2(x^2)

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suter [353]

Answer:

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b) 2502 miles

Step-by-step explanation:

You are given a number of miles and a number of gallons for the car. The ratio of these numbers is ...

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We presume this value is constant for this car. Then the questions can be answered by forming and solving a proportion:

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Multiplying by gallons and dividing by 18 mi/gal, we get

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It will take 91 1/6 gallons of gasoline for Mary to drive to her friend's house.

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b) Same deal, but the miles are unknown.

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3 years ago
Adult male heights have a normal probability distribution with a mean of 70 inches and a standard deviation of 4 inches. What is
raketka [301]

Answer: 0.5

Step-by-step explanation:

Given : Adult male heights have a normal probability distribution .

Population mean : \mu = 70 \text{ inches}

Standard deviation: \sigma= 4\text{ inches}

Let x be the random variable that represent the heights of adult male.

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For x=70, we have

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Now, by using the standard normal distribution table, we have

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Step-by-step explanation:

=x+3x+34x-34+48

=38x+14

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2 years ago
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