To solve this problem, we need to express the given function $y = (4x + 11)^4$ in terms of two functions: $y = f(u)$ and $u = g(x)$. After that, we will find the derivative $\frac{dy}{dx}$.

Step 1: Express the function in the form $y = f(u)$ and $u = g(x)$

Let:

• $u = g(x) = 4x + 11$
• Then, $y = f(u) = u^4$

Step 2: Differentiate using the chain rule

By the chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Step 3: Find $\frac{dy}{du}$ and $\frac{du}{dx}$

• Since $y = u^4$, we have: $\frac{dy}{du} = 4u^3$
• Since $u = 4x + 11$, we have: $\frac{du}{dx} = 4$

Step 4: Combine these results

Now, substituting into the chain rule: $\frac{dy}{dx} = 4u^3 \cdot 4 = 16u^3$

Step 5: Substitute back $u = 4x + 11$

Finally, replacing $u$ with $4x + 11$, we get: $\frac{dy}{dx} = 16(4x + 11)^3$

So, the derivative of $y = (4x + 11)^4$ with respect to $x$ is: $\frac{dy}{dx} = 16(4x + 11)^3$

Do you want more details or have any questions?

Here are 5 related questions for further practice:

1. How do you apply the chain rule to a composition of three functions?
2. What happens if the exponent in the original function is negative?
3. How do you find the second derivative of this function?
4. How does the chain rule change if the inner function is more complex?
5. Can you apply the product rule and chain rule together in other problems?

Tip: The chain rule is particularly useful for composite functions, so recognizing when one function is inside another is crucial to applying it correctly.