https://file.aiquickdraw.com/mathbot/file/17261072162936tr0tso6.jpg

The expression provided in the image involves a quotient, and you're asked to differentiate it using the quotient rule. The given function is:

\[
\frac{ax + b}{a^2 x^2 + b^2}
\]

We will differentiate this function with respect to \(x\) using the quotient rule.

### Quotient Rule:
If you have a function \(\frac{f(x)}{g(x)}\), its derivative is:

\[
\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}
\]

In this case, we have:
- \(f(x) = ax + b\)
- \(g(x) = a^2 x^2 + b^2\)

Let's compute the derivatives of \(f(x)\) and \(g(x)\).

1. **\(f(x) = ax + b\):**
   \[
   f'(x) = a
   \]

2. **\(g(x) = a^2 x^2 + b^2\):**
   \[
   g'(x) = 2a^2 x
   \]

Now, applying the quotient rule:

\[
\frac{d}{dx}\left(\frac{ax + b}{a^2 x^2 + b^2}\right) = \frac{(a)(a^2 x^2 + b^2) - (ax + b)(2a^2 x)}{(a^2 x^2 + b^2)^2}
\]

Let's simplify the numerator step by step:

\[
= a(a^2 x^2 + b^2) - (ax + b)(2a^2 x)
\]
\[
= a^3 x^2 + ab^2 - 2a^3 x^2 - 2a^2 bx
\]

Combining like terms:

\[
= -a^3 x^2 + ab^2 - 2a^2 bx
\]

Thus, the derivative is:

\[
\frac{-a^3 x^2 + ab^2 - 2a^2 bx}{(a^2 x^2 + b^2)^2}
\]

This is the simplified expression for the derivative.

Would you like a more detailed breakdown, or have any questions on specific steps?

Here are 5 related questions to practice:
1. What is the product rule for differentiation?
2. How does the quotient rule apply to trigonometric functions?
3. What happens if \( b = 0 \) in the given problem?
4. How can you apply the chain rule to composite functions?
5. What is the physical meaning of the derivative of a quotient in physics or economics?

**Tip:** Always simplify the numerator and denominator separately before dividing when applying the quotient rule to avoid errors.

Differentiate the function (ax + b) / (a^2 x^2 + b^2) using the quotient rule.

This article covers the detailed step-by-step differentiation of the function (ax + b) / (a^2 x^2 + b^2) using the quotient rule. The solution involves applying the quotient rule, simplifying the numerator, and finalizing the derivative expression. Ideal for university-level calculus students.

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