https://file.aiquickdraw.com/mathbot/file/1726306954849injoim6z.jpg

Let's analyze the problem.

We are asked to find \(\frac{dy}{dx}\) using the Chain Rule for the given function:
\[
y = \sqrt{49 + 343 \tan(6x)}
\]
at the point \(x = 0\).

### Step 1: Rewrite the function
The given function is:
\[
y = \sqrt{49 + 343 \tan(6x)} = (49 + 343 \tan(6x))^{1/2}
\]

### Step 2: Differentiate using the Chain Rule
We'll apply the Chain Rule, which states:
\[
\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)
\]
Here, let:
\[
f(u) = u^{1/2} \quad \text{where} \quad u = 49 + 343 \tan(6x)
\]

First, differentiate \(f(u)\) with respect to \(u\):
\[
f'(u) = \frac{1}{2} u^{-1/2}
\]

Next, differentiate \(u = 49 + 343 \tan(6x)\) with respect to \(x\):
\[
\frac{du}{dx} = 343 \cdot \sec^2(6x) \cdot 6 = 2058 \sec^2(6x)
\]

Thus, by the Chain Rule:
\[
\frac{dy}{dx} = \frac{1}{2}(49 + 343 \tan(6x))^{-1/2} \cdot 2058 \sec^2(6x)
\]

### Step 3: Evaluate at \(x = 0\)
At \(x = 0\), we have:
\[
\tan(6 \cdot 0) = 0 \quad \text{and} \quad \sec(6 \cdot 0) = 1
\]

So, the expression for \(u\) at \(x = 0\) is:
\[
u = 49 + 343 \cdot 0 = 49
\]
Therefore:
\[
\frac{dy}{dx} = \frac{1}{2} \cdot 49^{-1/2} \cdot 2058 \cdot 1^2
\]

\[
\frac{dy}{dx} = \frac{1}{2} \cdot \frac{2058}{7} = \frac{2058}{14} = 147
\]

### Final Answer:
The value of \(\frac{dy}{dx}\) at \(x = 0\) is \(147\).

Would you like further details on any of the steps or the Chain Rule?

Here are 5 related questions:
1. How does the Chain Rule work for compositions of more than two functions?
2. What is the derivative of \(\tan(x)\) and how does it apply here?
3. How would the problem change if the function involved \(\cot(x)\) instead of \(\tan(x)\)?
4. What role does the constant \(343\) play in the differentiation process?
5. How would you compute the second derivative for this function?

**Tip**: Always check if the function simplifies at the point you're evaluating before differentiating—it can save time and reduce errors.

Using the Chain Rule, find the value of dy/dx at the point x=0 where y = sqrt(49 + 343tan(6x)).

Find the derivative of y = sqrt(49 + 343tan(6x)) at the point x = 0 using the Chain Rule. This problem covers key differentiation concepts, trigonometric functions, and involves evaluating derivatives at specific points. Suitable for college-level calculus students.

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