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Katarina [22]
2 years ago
13

54. Find

ddle" class="latex-formula"> if x^y=y^x\text{.}
Mathematics
1 answer:
Stella [2.4K]2 years ago
7 0

Step-by-step explanation:

Take the natural log of both sides:

ln ({x}^{y} ) = ln ({y}^{x} )

Logarithm rules allow you to bring down the exponents:

yln(x) = xln(y)

Now differentiate. We will have to implicitly differentiate 'y' since it is a function of 'x'. Both sides require the product rule:

\frac{dy}{dx} ln(x) +  \frac{y}{x}  = ln(y) +  \frac{x}{y}  \frac{dy}{dx}

Isolate the terms that have y' since that is what we want:

\frac{dy}{dx} ln(x) -  \frac{x}{y}  \frac{dy}{dx}= ln(y) - \frac{y}{x}

Factor out y' to get:

\frac{dy}{dx}( ln(x) -   \frac{x}{y})= ln(y) - \frac{y}{x}

Therefore:

\frac{dy}{dx} =  \frac{ln(y) -  \frac{y}{x} }{ln(x) -  \frac{x}{y} }

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Does any one know how to do this. It would be such a life saver if you could help me out. Thx
Aneli [31]

Answer:

1: (3 ,- 4)

2:up (a > o)

3: -4

4: 2 units to the right

5: 4 units down

6: y = (x-2)^2-4

7:

8:

9: (5,9)

10: x=5

11: (4,7) and (6,7)

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Step-by-step explanation:

I'm not sure if you wanted it explained or answered , so i just answered most of them. Sorry if you wanted it explained.

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Answer:

C

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3x + 3y = -x + 5y

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2(1)=2 is a true statement

(2,4) when x=2, y=4

2(2)=4 is a true statement

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