Differentiate d(l-x÷v +n(x²+a²)^½÷v)÷dx =0 and find x

Given the function \( f(x) = l - \frac{x}{v} + \frac{n \sqrt{x^2 + a^2}}{v} \), we need to differentiate \( f(x) \) with respect to \( x \) and then find \( x \) such that the derivative is equal to zero.

### Step 1: Differentiate the function
Let's differentiate each term with respect to \( x \):

1. The first term \( l \) is a constant, so its derivative is 0.
2. The second term \( -\frac{x}{v} \) differentiates as:
   \[
   \frac{d}{dx}\left(-\frac{x}{v}\right) = -\frac{1}{v}
   \]
3. The third term \( \frac{n \sqrt{x^2 + a^2}}{v} \) differentiates as follows:
   \[
   \frac{d}{dx} \left( \frac{n \sqrt{x^2 + a^2}}{v} \right) = \frac{n}{v} \cdot \frac{d}{dx}\left( \sqrt{x^2 + a^2} \right)
   \]
   The derivative of \( \sqrt{x^2 + a^2} \) with respect to \( x \) is:
   \[
   \frac{d}{dx}\left( \sqrt{x^2 + a^2} \right) = \frac{1}{2\sqrt{x^2 + a^2}} \cdot 2x = \frac{x}{\sqrt{x^2 + a^2}}
   \]
   Therefore, the derivative of the third term is:
   \[
   \frac{n}{v} \cdot \frac{x}{\sqrt{x^2 + a^2}}
   \]

### Step 2: Combine the derivatives
Combining the results from the differentiation steps above, we have:
\[
\frac{d}{dx} f(x) = -\frac{1}{v} + \frac{n}{v} \cdot \frac{x}{\sqrt{x^2 + a^2}}
\]

### Step 3: Set the derivative equal to zero
To find the critical points, set the derivative equal to zero:
\[
-\frac{1}{v} + \frac{n}{v} \cdot \frac{x}{\sqrt{x^2 + a^2}} = 0
\]
Simplifying, we get:
\[
\frac{n}{v} \cdot \frac{x}{\sqrt{x^2 + a^2}} = \frac{1}{v}
\]
\[
n \cdot \frac{x}{\sqrt{x^2 + a^2}} = 1
\]
\[
\frac{x}{\sqrt{x^2 + a^2}} = \frac{1}{n}
\]

### Step 4: Solve for \( x \)
Square both sides to eliminate the square root:
\[
\frac{x^2}{x^2 + a^2} = \frac{1}{n^2}
\]
Cross-multiplying gives:
\[
x^2 = \frac{x^2 + a^2}{n^2}
\]
\[
n^2 x^2 = x^2 + a^2
\]
\[
(n^2 - 1)x^2 = a^2
\]
\[
x^2 = \frac{a^2}{n^2 - 1}
\]
Taking the square root of both sides:
\[
x = \pm \frac{a}{\sqrt{n^2 - 1}}
\]

### Final Answer:
The solutions for \( x \) are:
\[
x = \pm \frac{a}{\sqrt{n^2 - 1}}
\]

Would you like more details or have any questions?

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**Related Questions:**
1. How does differentiation help in finding critical points of a function?
2. What are the implications of a constant term in differentiation?
3. How do square roots and square terms interact in differentiation?
4. Can \( n \) be any value, or are there restrictions?
5. What happens if \( n^2 - 1 \leq 0 \)?

**Tip:** When dealing with square roots in differentiation, always check for potential restrictions in the domain of the function.

Learn how to differentiate the equation d(l-x÷v +n(x²+a²)^½÷v)÷dx =0 step-by-step. Understand the application of differentiation rules and square root derivatives in solving mathematical equations. Suitable for advanced high school students and college-level learners.

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