Let's solve both problems using the chain rule.

Problem 13:

Given: $y = \frac{2(x + 4)}{(x + 4)^2 - 1}$ To differentiate this, we'll apply the quotient rule along with the chain rule.

Step-by-step differentiation:

Let: $u = 2(x + 4), \quad v = (x + 4)^2 - 1$ The quotient rule states: $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}$

Now we calculate $\frac{du}{dx}$ and $\frac{dv}{dx}$.

1. $\frac{du}{dx} = \frac{d}{dx} \left( 2(x + 4) \right) = 2$
2. $\frac{dv}{dx} = \frac{d}{dx} \left( (x + 4)^2 - 1 \right) = 2(x + 4)$

Now, applying the quotient rule: $\frac{dy}{dx} = \frac{\left( (x + 4)^2 - 1 \right)(2) - \left( 2(x + 4) \right)(2(x + 4))}{\left( (x + 4)^2 - 1 \right)^2}$ Simplify: $\frac{dy}{dx} = \frac{2 \left( (x + 4)^2 - 1 \right) - 4(x + 4)^2}{\left( (x + 4)^2 - 1 \right)^2}$ This can be further simplified, but we'll leave it here for now.

Problem 14:

Given: $y = \frac{1}{(1 + \sqrt{1 - x})^2}$ This requires the chain rule because of the square and the square root.

Step-by-step differentiation:

Rewrite the function as: $y = (1 + \sqrt{1 - x})^{-2}$

Now, apply the chain rule.

1. First, let $u = 1 + \sqrt{1 - x}$. Then $y = u^{-2}$.

2. Differentiate $y = u^{-2}$ with respect to $u$: $\frac{dy}{du} = -2u^{-3}$

3. Now differentiate $u = 1 + \sqrt{1 - x}$ with respect to $x$: $\frac{du}{dx} = \frac{d}{dx} \left( 1 + (1 - x)^{1/2} \right) = \frac{1}{2}(1 - x)^{-1/2} \cdot (-1) = \frac{-1}{2\sqrt{1 - x}}$

Now, combine the two results using the chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -2u^{-3} \cdot \frac{-1}{2\sqrt{1 - x}}$

Substitute $u = 1 + \sqrt{1 - x}$ back into the expression: $\frac{dy}{dx} = \frac{2}{(1 + \sqrt{1 - x})^3 \cdot 2\sqrt{1 - x}}$ This simplifies to: $\frac{dy}{dx} = \frac{1}{(1 + \sqrt{1 - x})^3 \cdot \sqrt{1 - x}}$

Would you like a more detailed step in any part?

Here are some related questions you might want to explore:

1. How do you simplify expressions after applying the quotient rule?
2. What is the best strategy for differentiating complicated functions involving fractions?
3. Can the chain rule be combined with implicit differentiation?
4. How can we handle higher-order derivatives of these functions?
5. What role do critical points play in determining the behavior of these functions?

Tip: When using the chain rule, carefully identify each "inner" and "outer" function to differentiate correctly.