Respuesta :

xKelvin

Answer:

A

Step-by-step explanation:

We are given the equation:  

[tex]x^2+xy-3y=3[/tex]

And we want to find dy/dx at the point (2, 1).

Find dy/dx. We can take the derivative of both sides with respect to x:

[tex]\displaystyle \frac{d}{dx}[x^2+xy-3y]=\frac{d}{dx}[3][/tex]

Implicitly differentiate:

[tex]\displaystyle 2x+y+x\frac{dy}{dx}-3\frac{dy}{dx}=0[/tex]

Solve for dy/dx:

[tex]\displaystyle \begin{aligned} x \frac{dy}{dx} - 3\frac{dy}{dx} & = -2x - y \\ \\ \frac{dy}{dx}(x - 3) & = -2x - y \\ \\ \frac{dy}{dx} & = \frac{-2x-y}{x-3} \\ \\ & = \frac{2x+y}{3-x} \end{aligned}[/tex]

To find dy/dx at (2, 1), evaluate dy/dx for x = 2 and y = 1. Hence:

[tex]\displaystyle \frac{dy}{dx}\Big|_{(2, 1)}=\frac{2(2)+1}{3 - (2)}=\frac{5}{1}=5[/tex]

Hence, our answer is A.

facundo3141592

We want to find the differential dy/dx for the given expression at the given point. We will see that the correct option is A.

How to differentiate the expression?

We have:

x^2 + xy - 3y = 3

First, we need to isolate the variable y in one side of the equation, so we get:

xy - 3y = 3 - x^2

y*(x - 3) = 3 - x^2

y = (x - x^2)/(x - 3)

Now we can differentiate this, remember that if:

f(x) = g(x)*h(x)

then:

f'(x) = g(x)*h'(x) + g'(x)*h'(x)

Using that rule.

dy/dx = (1 - 2x)/(x - 3) - (x - x^2)/(x - 3)^2

Now we need to evaluate this in x = 2 (the x-value of the point) so we get:

(1 - 2*2)/(2 - 3) - (2 - 2^2)/(2 - 3)^2

= (-3)/(-1) - (-2)/(1) = 5

So the correct option is A.

If you want to learn more about differentiation, you can read:

https://brainly.com/question/24898810

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