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lara31 [8.8K]
4 years ago
6

Let f(X) = (x^7(x-6)^8)/(x^2+5)^9

Mathematics
1 answer:
Semenov [28]4 years ago
5 0
First a review of log properties:
log (ab) = log(a) + log(b)
log (a/b) = log(a) - log(b)
log (a^n) = n*log(a)

In this way by taking the log of entire function we can break it up into more simple terms before trying to take the derivative.

But we must take log of BOTH sides.
y = \frac{x^7 (x-6)^8}{(x^2+5)^9}   \\ ln (y) = ln (\frac{x^7 (x-6)^8}{(x^2+5)^9})
Expand Right side
ln(y) = ln (x^7) + ln (x-6)^8 - ln (x^2+5)^9  \\ ln (y) = 7 ln (x) +8 ln (x-6) - 9 ln(x^2 +5)
Now lets review derivative of natural log:
\frac{d}{dx} ln( u(x)) = \frac{1}{u(x)} \frac{du}{dx}
Take derivative of both sides:
\frac{1}{y} \frac{dy}{dx} = 7(\frac{1}{x})(1) + 8(\frac{1}{x-6})(1) - 9(\frac{1}{x^2+5})(2x)  \\ \frac{1}{y} \frac{dy}{dx} = \frac{7}{x} + \frac{8}{x-6} - \frac{18x}{x^2+5}
Solve for dy/dx by multiplying by "y". (Note y is original function)
\frac{dy}{dx} = y(\frac{7}{x} + \frac{8}{x-6} - \frac{18x}{x^2+5}) \\ \frac{dy}{dx} = (\frac{x^7 (x-6)^8}{(x^2+5)^9})(\frac{7}{x} + \frac{8}{x-6} - \frac{18x}{x^2+5})
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3 years ago
Find the distance between each pair of points. Round to the nearest tenth.
Vsevolod [243]

Answer:

The answer is

<h2>6.7 units</h2>

Step-by-step explanation:

The distance between two points can be found by using the formula

d =  \sqrt{ ({x1 - x2})^{2} +  ({y1 - y2})^{2}  } \\

where

(x1 , y1) and (x2 , y2) are the points

From the question the points are

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The distance between them is

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We have the final answer as

<h3>6.7 units to the nearest tenth</h3>

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I will assume this is a right triangle with the information given

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