2y^3 - 3xy = 4
1) Find dy/dx
2) Write an equation for the line tangent to the curve at (1,2)
3) Find d^2y/dx^2 at (1,2)
I answered this already, in considerable detail. If you have changed your screen name from "Karen" and resubmitted the same question, look elsewhere. If not, review my answer at
http://www.jiskha.com/display.cgi?id=1229810620
and show your work.
o yeah, sry, i was accidentally on my sister's thing when i asked first
To find the derivative dy/dx, we need to use the rules of differentiation. In this case, we have a function in two variables, y and x, so we'll use the partial derivative with respect to x.
1) Find dy/dx:
To differentiate 2y^3 - 3xy = 4 with respect to x, we consider y as a function of x and differentiate each term while treating y as the independent variable. The derivative of y with respect to x is denoted as dy/dx. Let's differentiate each term:
The derivative of 2y^3 with respect to x is (2 * 3y^2) * (dy/dx) = 6y^2 * (dy/dx)
The derivative of -3xy with respect to x is -3y - 3x * (dy/dx)
Since these two terms form the equation, their sum must be equal to zero:
6y^2 * (dy/dx) + (-3y - 3x * (dy/dx)) = 0
Now, let's isolate dy/dx:
6y^2 * (dy/dx) - 3y - 3x * (dy/dx) = 0
(6y^2 - 3x) * (dy/dx) = 3y
dy/dx = 3y / (6y^2 - 3x)
2) Write the equation for the line tangent to the curve at (1,2):
To find the equation of the tangent line at (1,2), we need both the slope (given by dy/dx) and the point (1,2). We already found dy/dx, so we substitute the values into the point-slope form of a line, y - y1 = m(x - x1):
Using (1,2):
x1 = 1
y1 = 2
Substituting dy/dx = 3y / (6y^2 - 3x) and the given point (1,2) into the equation:
y - 2 = (3(2)) / (6(2)^2 - 3(1))
y - 2 = 6 / (24 - 3)
y - 2 = 6 / 21
Simplifying, we get:
y - 2 = 2 / 7
So the equation of the line tangent to the curve at (1,2) is:
y = (2/7) + 2
3) Find d^2y/dx^2 at (1,2):
To find the second derivative, we need to differentiate the derivative from step 1 with respect to x:
d/dx [dy/dx] = d/dx [3y / (6y^2 - 3x)]
To differentiate this expression, we apply the quotient rule:
d/dx [3y / (6y^2 - 3x)] = (3 * (6y^2 - 3x) * d/dx [y] - 3y * d/dx [6y^2 - 3x]) / (6y^2 - 3x)^2
Now, let's calculate d^2y/dx^2 at the point (1,2) by substituting x = 1 and y = 2 into the expression:
(3 * (6(2)^2 - 3(1)) * d/dx [y] - 3(2) * d/dx [6(2)^2 - 3(1)]) / (6(2)^2 - 3(1))^2
(3 * (24 - 3) * d/dx [y] - 3(2) * d/dx [24 - 3]) / (24 - 3)^2
Simplifying the expression yields:
(3 * 21 * d/dx [y] - 6 * d/dx [21]) / 21^2
Since we don't have further information about d/dx [y] and d/dx [21], we can't evaluate d^2y/dx^2 at (1,2).