Using 4x^2 - 9y^2 = 36

Find y by implicit differentiation and then solve the equation explicitly for y and differentiate to get y' in terms of x.

To find y by implicit differentiation, we differentiate both sides of the equation with respect to x. Let me explain the process step-by-step:

Step 1: Differentiate each term on the left side of the equation with respect to x.
The derivative of 4x^2 with respect to x is 8x.
The derivative of -9y^2 with respect to x is -18yy'.

Step 2: Differentiate the constant term on the right side of the equation.
The derivative of 36 with respect to x is 0.

So, after differentiating both sides of the equation, we have:
8x - 18yy' = 0

Now let's solve this equation explicitly for y by isolating y:

Step 1: Move the term containing y' to the other side of the equation.
-18yy' = -8x

Step 2: Divide both sides of the equation by -18y to solve for y.
y' = -8x / (-18y)

Simplifying this equation gives us:
y' = 4x / (9y)

Hence, y is explicitly given by:
y = (4x) / (9y)

And y' in terms of x is given by:
y' = 4x / (9y)

Therefore, the solution for y and y' in terms of x, obtained by using implicit differentiation, is y = (4x) / (9y) and y' = 4x / (9y).