Question

Find y ' by implicit differentiation. Simplify where possible. x² - 3y² = 3.

Answer 1

Implicit differentiation is used to find the derivative of a function when it is difficult to solve explicitly for y in terms of x. In this case, we differentiate each term of the equation x² - 3y² = 3 and solve for y' to find the derivative.

Step-by-step explanation:

To find the derivative of y with respect to x by implicit differentiation, we will treat y as a function of x and differentiate both sides of the equation.

Starting with the given equation x² - 3y² = 3, we differentiate each term with respect to x.

d/dx(x²) - d/dx(3y²) = d/dx(3)

Simplifying, we get 2x - 6yy' = 0.

Now, we can solve for y' by isolating it: y' = 2x / (6y).

Therefore, the derivative of y with respect to x is y' = 2x / (6y).

Answer 2

To find the derivative y' of the function defined implicitly by x² - 3y² = 3, we differentiate each term with respect to x, solve for y', and simplify to get y' = ⅓x/y.

Step-by-step explanation:

To find y' by implicit differentiation for the equation x² - 3y² = 3, follow these steps:

1. Differentiate both sides of the equation with respect to x. Remember to use the chain rule for the y² term.
2. For the x² term, the derivative with respect to x is 2x. For the -3y² terms, the derivative is -6yy' (since y is a function of x).
3. Setting the derivatives equal yields 2x - 6yy' = 0.
4. Solve this equation for y' to find y' = ⅓x/y.

Thus, the derivative of y with respect to x, or y', is ⅓x/y.