In the image, you are asked to express the given function $y = \sqrt{x^3 + 4}$ as a composite function in the form $f(g(x))$, identify the inner function $u = g(x)$, and the outer function $y = f(u)$, and then find the derivative $\frac{dy}{dx}$.

Step 1: Express as a Composite Function $f(g(x))$

We are given $y = \sqrt{x^3 + 4}$. We can break this down into two functions:

• The inner function $g(x)$ is the expression inside the square root:
  $g(x) = x^3 + 4$
• The outer function $f(u)$ takes the square root of $u$:
  $f(u) = \sqrt{u}$

Thus, the composite function can be written as: $y = f(g(x)) = \sqrt{x^3 + 4}$

Step 2: Find the Derivative $\frac{dy}{dx}$

To differentiate $y = \sqrt{x^3 + 4}$, we'll use the chain rule.

$\frac{dy}{dx} = \frac{df}{du} \cdot \frac{dg}{dx}$

• First, differentiate the outer function $f(u) = \sqrt{u}$: $\frac{df}{du} = \frac{1}{2\sqrt{u}} = \frac{1}{2\sqrt{x^3 + 4}}$

• Next, differentiate the inner function $g(x) = x^3 + 4$: $\frac{dg}{dx} = 3x^2$

Now, applying the chain rule: $\frac{dy}{dx} = \frac{1}{2\sqrt{x^3 + 4}} \cdot 3x^2 = \frac{3x^2}{2\sqrt{x^3 + 4}}$

So, the derivative is: $\frac{dy}{dx} = \frac{3x^2}{2\sqrt{x^3 + 4}}$

Would you like to dive deeper into any of these steps? Let me know if you have more questions!

Here are 5 related questions to expand your understanding:

1. How does the chain rule work for functions with more than two nested functions?
2. What is the importance of identifying the inner and outer functions in a composite function?
3. How do you compute higher-order derivatives of composite functions?
4. What other methods can be used to differentiate complex functions besides the chain rule?
5. How would the result change if the original function involved a different power, such as $x^5$, instead of $x^3$?

Tip:

Always check the domain of composite functions, especially when dealing with square roots, to ensure the expression inside is non-negative.