find the first and second derivatives for 3x^4-4xy^2+7=2y
3x^4-4xy^2+7=2y
12x^3 - 4y^2 - 8xyy' = 2y'
(8xy+2)y' = 12x^3-4y^2
y' = (6x^3-3y^2)/(4xy+1)
y" =
(18x^2-6yy')(4xy+1) - (6x^3-3y^2)(4y+4xy')
-------------------------------------------------------
(4xy+1)
Now it's just algebra. Substitute the value of y' in the above expression and then massage it as you will.
oops. The denominator is (4xy+1)^2
See if you can get what wolframalpha.com gets:
http://www.wolframalpha.com/input/?i=%28%2818x^2-6y%28%286x^3-3y^2%29%2F%284xy%2B1%29%29%29%284xy%2B1%29+-+%286x^3-3y^2%29%284y%2B4x%28%286x^3-3y^2%29%2F%284xy%2B1%29%29%29+%29%2F%284xy%2B1%29+^2
To find the first and second derivatives of the given function, we can use the rules of differentiation. Let's start with the first derivative:
1. First, let's rewrite the equation as a function: f(x, y) = 3x^4 - 4xy^2 + 7 - 2y.
2. To find the partial derivative with respect to x, we treat y as a constant and differentiate only with respect to x. So, we differentiate each term of the function with respect to x:
∂/∂x (3x^4) = 12x^3
∂/∂x (-4xy^2) = -4y^2
∂/∂x (7) = 0
∂/∂x (-2y) = 0
Therefore, the partial derivative of f(x, y) with respect to x (df/dx) is: df/dx = 12x^3 - 4y^2.
3. Now, let's find the partial derivative with respect to y. Similarly, we treat x as a constant and differentiate with respect to y:
∂/∂y (3x^4) = 0
∂/∂y (-4xy^2) = -8xy
∂/∂y (7) = 0
∂/∂y (-2y) = -2
Therefore, the partial derivative of f(x, y) with respect to y (df/dy) is: df/dy = -8xy - 2.
Now, let's proceed to find the second derivatives:
4. To find the second derivative with respect to x (d²f/dx²), we differentiate the first derivative (df/dx) with respect to x:
d²f/dx² = d/dx (12x^3 - 4y^2)
= 36x^2 - 0
= 36x^2
5. Similarly, to find the second derivative with respect to y (d²f/dy²), we differentiate the first derivative (df/dy) with respect to y:
d²f/dy² = d/dy (-8xy - 2)
= -8x - 0
= -8x
Therefore, the first derivative is df/dx = 12x^3 - 4y^2, df/dy = -8xy - 2, and the second derivatives are d²f/dx² = 36x^2 and d²f/dy² = -8x.