Final answer:
To find y' by implicit differentiation, we differentiate both sides of the equation. The derivative of 6x^2 with respect to x is 12x, and the derivative of -y^2 with respect to x is -2yy'. Setting these equal to 0, we obtain 12x - 2yy' = 0. Rearranging this equation, we find y' = -6x/y.
Step-by-step explanation:
To find y' by implicit differentiation, we differentiate both sides of the equation with respect to x. Let's start:
6x2 - y2 = 3
Now, differentiate both sides:
- Using the power rule, the derivative of 6x2 with respect to x is 12x.
- For the derivative of -y2 with respect to x, we use the chain rule. The derivative of -y2 with respect to y is -2y, and since y is a function of x, we multiply by the derivative of y with respect to x, which is y'.
- The derivative of 3 with respect to x is 0, as it is a constant.
Combining these results, we have:
12x - 2yy' = 0
To solve the equation explicitly for y and differentiate to get y' in terms of x, we rearrange the equation as follows:
- Subtract 12x from both sides: -2yy' = -12x
- Divide all terms by -2y to isolate y': y' = -6x/y