For the equation given below, evaluate dy/dx 3y^2-3x^2+9=0 at the point (2,1)
To evaluate dy/dx for the equation 3y^2 - 3x^2 + 9 = 0 at the point (2, 1), we can use implicit differentiation.
1. Start by differentiating both sides of the equation with respect to x:
d/dx (3y^2) - d/dx (3x^2) + d/dx(9) = d/dx(0)
2. Apply the chain rule to differentiate y^2 with respect to x:
6y * dy/dx - 6x * dx/dx + 0 = 0 (since dx/dx = 1 and d/dx(9) = 0)
3. Simplify the equation by substituting the given values of (2, 1):
6(1) * dy/dx - 6(2) * 1 + 0 = 0
4. Further simplify the equation:
6 * dy/dx - 12 = 0
5. Solve for dy/dx:
6 * dy/dx = 12
Divide both sides by 6:
dy/dx = 12/6
Simplify:
dy/dx = 2
So, evaluate dy/dx for the equation 3y^2 - 3x^2 + 9 = 0 at the point (2, 1) is equal to 2.
To evaluate dy/dx for the given equation at the point (2,1), we can first rewrite the equation in terms of y as follows:
3y^2 - 3x^2 + 9 = 0
Next, we differentiate both sides of the equation with respect to x using the chain rule.
To differentiate y^2 with respect to x, we can use the chain rule by treating y as a function of x. So, we have:
d/dx (y^2) = d/dy (y^2) * dy/dx
Similarly, differentiating -3x^2 and 9 with respect to x, we get:
d/dx (-3x^2) = -6x
d/dx (9) = 0
Plugging these derivatives back into the equation, we have:
2 * 3y * dy/dx - 6x + 0 = 0
Now, substitute the given point (2,1) into the equation. So, x = 2 and y = 1:
2 * 3(1) * dy/dx - 6(2) = 0
6 * dy/dx - 12 = 0
Simplify the equation:
6 * dy/dx = 12
Now, solve for dy/dx:
dy/dx = 12/6 = 2
Therefore, the value of dy/dx for the given equation at the point (2,1) is 2.