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Schach [20]
2 years ago
7

Find dy/dx if 3xy=4x+y^2

Advanced Placement (AP)
1 answer:
ElenaW [278]2 years ago
3 0

Hi there!

We can use implicit differentiation with respect to x:

3xy = 4x + y^2

If a term with 'y' is differentiated, a 'dy/dx' must be included.

We can differentiate each term separately for the explanation.

<u>3xy</u>

We must use the power rule since 'x' and 'y' are both in this term.

Power rule:

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

3xy \\\\f(x) = 3x\\g(x) = y \\\\f'(x)g(x) + g'(x)f(x) = 3y + 3x\frac{dy}{dx}

Now, we can do the others.

<u>4x</u>

This is a normal power rule derivative.

f(x) = 4x\\f'(x) = 4

<u>y²</u>

Since we are not differentiating with respect to y, we must include 'dy/dx'.

f(x) = y^2\\f'(x) = 2y\frac{dy}{dx}

Combine the above:

3y + 3x\frac{dy}{dx} = 4 + 2y\frac{dy}{dx}

Rearrange to solve for dy/dx.

Move dy/dx to one side:

3x\frac{dy}{dx} - 2y\frac{dy}{dx} = 4 + 3y \\\\

Factor out dy/dx and divide:

\frac{dy}{dx}(3x- 2y) = 4 + 3y \\\\\boxed{\frac{dy}{dx} = \frac{4+3y}{3x-2y}}

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