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The question asks for the derivative of the function \( \tan^{-1}(7x) \) with respect to \( x \).

### Solution:

Given: \( f(x) = \tan^{-1}(7x) \)

We will apply the chain rule for differentiation, where the derivative of \( \tan^{-1}(u) \) with respect to \( u \) is:

\[
\frac{d}{dx} \left( \tan^{-1}(u) \right) = \frac{1}{1 + u^2}
\]

Here, \( u = 7x \). Now applying the chain rule:

\[
\frac{d}{dx} \left( \tan^{-1}(7x) \right) = \frac{1}{1 + (7x)^2} \cdot \frac{d}{dx}(7x)
\]

The derivative of \( 7x \) with respect to \( x \) is 7. So:

\[
\frac{d}{dx} \left( \tan^{-1}(7x) \right) = \frac{7}{1 + 49x^2}
\]

### Answer:
The correct option is \( \boxed{B} \).

Would you like further clarification or more details on this solution?

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Here are 5 related questions:
1. What is the derivative of \( \tan^{-1}(x) \)?
2. How would you apply the chain rule to other inverse trigonometric functions?
3. Can you find the derivative of \( \tan^{-1}(kx) \) for any constant \( k \)?
4. How does the chain rule affect the differentiation of composite functions?
5. What is the general derivative of \( \tan^{-1}(ax + b) \)?

**Tip:** Remember, the chain rule helps differentiate composite functions by multiplying the outer function's derivative with the inner function's derivative.

x এর সাপেক্ষে tan⁻¹ 7x এর অন্তরজ:

Find the derivative of tan⁻¹(7x) using the chain rule for differentiation. Learn about the application of the inverse trigonometric differentiation formula and the chain rule. This problem is suitable for advanced high school or undergraduate calculus students.

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