Find dy/dx at the point (3,8) for 5xy^2/3-x^2y=-12
A.2
B.17
C.-7
D.-28/3
What was your dy/dx ?
5 x y^(2/3) - x^2 y= -12 ???
5[x y^-(1/3) (2/3) dy + y^(2/3)dx] = x^2 dy + 2 x y dx
now plug in x = 3 and y = 8
5[ 1 dy + 4 dx ] = 9 dy -20 dx
5 dy - 9 dy = 48dx - 20 dx
-4 dy = -28 dx
dy/dx = -28/4 = -7
I had to change the problem statement typing, figured it meant y^(2/3) not y^2 / 2
To find dy/dx at the point (3, 8), we can use implicit differentiation. Implicit differentiation allows us to find the derivative of y with respect to x even when y is not explicitly expressed in terms of x.
Let's start by differentiating both sides of the equation with respect to x:
For the left side, we need to use the product rule. The product rule states that if we have two functions u(x) and v(x), then the derivative of their product, u(x) * v(x), is given by:
(d/dx)[u(x) * v(x)] = u'(x) * v(x) + u(x) * v'(x)
For the right side, we differentiate a constant, which is 0.
Differentiating the left side, we have:
d/dx [5xy^(2/3) - x^2y] = 0.
Using the product rule, we can differentiate each term separately:
d/dx [5xy^(2/3)] - d/dx [x^2y] = 0.
For the first term, we need to use the chain rule. The chain rule states that if we have a composite function, for example, f(g(x)), then the derivative of this composite function is given by:
(d/dx) [f(g(x))] = f'(g(x)) * g'(x).
Differentiating the first term, we have:
5 * d/dx [xy^(2/3)] - d/dx [x^2y] = 0.
Differentiating each term separately:
5 * [y^(2/3) * d/dx(x)] + x * d/dx [y^(2/3)] - [2x * y + x^2 * d/dx(y)] = 0.
Since we want to find dy/dx at the point (3, 8), we can substitute these values into the equation.
Plugging in x = 3 and y = 8, we have:
5 * [8^(2/3) * d/dx(3)] + 3 * d/dx [8^(2/3)] - [2 * 3 * 8 + 3^2 * d/dx(8)] = 0.
Simplifying the equation:
5 * [8^(2/3) * d/dx(3)] + 3 * d/dx [8^(2/3)] - [48 + 9 * d/dx(8)] = 0.
Now, let's evaluate the derivatives and plug in the values:
5 * [8^(2/3) * 0] + 3 * 0 - [48 + 9 * 0] = 0.
Simplifying:
0 + 0 - [48 + 0] = 0.
Now, we have:
-48 = 0.
Since this equation is inconsistent, there is no specific value for dy/dx at the point (3, 8).
Therefore, none of the answer choices A, B, C, or D is correct.