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Sveta_85 [38]
3 years ago
9

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

Mathematics
1 answer:
Tasya [4]3 years ago
5 0

Answer:

\frac{dy}{dx}=-[(\frac{5x+24}{36x-6x^2})]

Step-by-step explanation:

Given function:

y =\ln(\frac{(6 - x)^{\frac{3}{2}}}{x^{\frac{2}{3}}})

we know

\ln(\frac{A}{B}) = ln(A) - ln(B)

thus,

y = \ln((6 - x)^{\frac{3}{2}}) - \ln(x^{\frac{2}{3}})

or

also,

ln(Aⁿ) = n × ln(A)

thus,

y = (\frac{3}{2})\times\ln(6 - x) - (\frac{2}{3})\times\ln(x)

therefore,

\frac{dy}{dx}=[(\frac{3}{2})\times\frac{1}{(6-x)}\times(0 - 1)] - [ (\frac{2}{3})\times\frac{1}{x}\times1]

or

\frac{dy}{dx}=-(\frac{3}{2(6-x)}) - (\frac{2}{3x})

or

\frac{dy}{dx}=-[(\frac{3(3x)+2\times2(6-x)}{2(6-x)\times(3x)})]

or

\frac{dy}{dx}=-[(\frac{5x+24}{36x-6x^2})]

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How many bucketloads would it take to bucket out the world’s oceans? Write your answer in scientific notation.
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Answer: 7\times10^{19} buckets

Step-by-step explanation:

Given:  A cubic kilometer=10^{15}cubic centimeters

The volume of  world’s oceans=1.4\times10^{9} cubic kilometers of water.

⇒ The volume of  world’s oceans=1.4\times10^{9}\times10^{15} cubic centimeters of water.

Volume of a bucket = 20,000 cubic centimeters of water.

The number of bucket-loads would it take to bucket out the world’s oceans

n=\frac{\text{volume of ocean}}{\text{volume of bucket}}=\frac{1.4\times10^{9}\times10^{15}}{20000}

\Rightarrow\ n=\frac{1.4\times10^{9+15}}{0.2\times10^5}......[a^n\times a^m=a^{m+n}]\\\Rightarrow\ n=7\times10^{24-5}.....[\frac{a^m}{a^n}=a^{m-n}]\\\Rightyarrow\ n=7\times10^{19}

hence,  7\times10^{19} bucketloads would it take to bucket out the world’s oceans.

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