Final answer:
To find y' via implicit differentiation of 9x² - y² = 7, differentiate both sides with respect to x, resulting in y' = -9x/y. Solving explicitly for y and then differentiating gives y' = ±9x/(sqrt(9x² - 7)).
Step-by-step explanation:
To find the derivative y' by implicit differentiation for the equation 9x² - y² = 7, we differentiate both sides with respect to x:
First, we differentiate the x-terms: d/dx (9x²) = 18x.
Next, we apply the chain rule to differentiate the y-term: d/dx (-y²) = -2yy', because y is a function of x.
So, we have 18x - 2yy' = 0.
To isolate y', divide both sides by -2y: y' = (-18x) / (2y) = -9x/y.
For part B, to solve the equation explicitly for y and differentiate to get y' in terms of x:
First, we express y in terms of x using the given equation: y² = 9x² - 7, so y = ±9sqrt(9x² - 7).
We then differentiate using the chain rule: y' = d/dx (±sqrt(9x² - 7)) = ±(1/2)(9x² - 7)^(-1/2)(18x) = ±9x/(sqrt(9x² - 7)).