Let's work through this problem step by step.

The equation given is:

$8x^2 - y^5 = 9$

(a) Find $y'$ by implicit differentiation.

1. Differentiate both sides of the equation with respect to $x$. For the left-hand side:
   • The derivative of $8x^2$ with respect to $x$ is $16x$.
   • The derivative of $y^5$ with respect to $x$ is $5y^4 \cdot \frac{dy}{dx}$ (since $y$ is a function of $x$).

So the equation becomes:

$16x - 5y^4 \cdot \frac{dy}{dx} = 0$

2. Now solve for $\frac{dy}{dx}$:

$16x = 5y^4 \cdot \frac{dy}{dx}$

$\frac{dy}{dx} = \frac{16x}{5y^4}$

Thus, $y' = \frac{16x}{5y^4}$.

(b) Solve the equation explicitly for $y$ and differentiate to get $y'$ in terms of $x$.

First, solve for $y^5$ in terms of $x$:

$y^5 = 8x^2 - 9$

Now solve for $y$:

$y = \left(8x^2 - 9\right)^{1/5}$

Next, differentiate this explicitly to find $y'$. Using the chain rule:

$y' = \frac{d}{dx} \left( \left(8x^2 - 9\right)^{1/5} \right)$

Applying the chain rule:

$y' = \frac{1}{5} \left(8x^2 - 9\right)^{-4/5} \cdot \frac{d}{dx} (8x^2 - 9)$

The derivative of $8x^2 - 9$ with respect to $x$ is $16x$, so:

$y' = \frac{1}{5} \left(8x^2 - 9\right)^{-4/5} \cdot 16x$

Thus:

$y' = \frac{16x}{5 \left(8x^2 - 9\right)^{4/5}}$

(c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for $y$ into your solution for part (a).

From part (a), we had:

$y' = \frac{16x}{5y^4}$

Substitute $y = \left(8x^2 - 9\right)^{1/5}$ into this equation. Since $y^4 = \left(8x^2 - 9\right)^{4/5}$, we get:

$y' = \frac{16x}{5 \left(8x^2 - 9\right)^{4/5}}$

This matches the result from part (b), confirming that both answers are consistent.

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. What is implicit differentiation, and how does it differ from explicit differentiation?
2. How can you apply the chain rule to differentiate composite functions?
3. Why is it important to check consistency between implicit and explicit differentiation?
4. How can you solve for higher-order derivatives in implicit differentiation problems?
5. How would you approach similar problems where the powers of $x$ and $y$ are more complex?

Tip: Always ensure that you properly apply the chain rule when differentiating implicitly, especially when dealing with higher powers of functions.