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2 years ago

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9. Scientists measured the spine size of a population of cactuses found in an area in 1970. They measured the spine size of the

cactus population again in 2015. Which of the statements below best describes the difference in the cactuses at the two time points?

Advanced Placement (AP)

1 answer:

ANEK [815]2 years ago

4 0

Answer: "There were more cactuses in 1970 than there were in 2015"

Explanation:

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3 years ago

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

ElenaW [278]

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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2 years ago