find the derivative of f(x)= -2 - (x/√(xˆ2+1))

To find the derivative of the function \( f(x) = -2 - \frac{x}{\sqrt{x^2 + 1}} \), we'll proceed with standard differentiation techniques, including the quotient rule and chain rule.

Given:
\[
f(x) = -2 - \frac{x}{\sqrt{x^2 + 1}}
\]

### Step 1: Differentiate the constant term
The derivative of \(-2\) is 0.

### Step 2: Differentiate the second term \(- \frac{x}{\sqrt{x^2 + 1}}\)
We'll use the quotient rule, which states:
\[
\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2}
\]
where \( u = x \) and \( v = \sqrt{x^2 + 1} \).

- First, compute \( u' \) (the derivative of the numerator):
\[
u' = \frac{d}{dx}(x) = 1
\]

- Next, compute \( v' \) (the derivative of the denominator):
\[
v = \sqrt{x^2 + 1} = (x^2 + 1)^{1/2}
\]
Using the chain rule:
\[
v' = \frac{1}{2}(x^2 + 1)^{-1/2} \cdot 2x = \frac{x}{\sqrt{x^2 + 1}}
\]

### Step 3: Apply the quotient rule
Now apply the quotient rule to differentiate \( -\frac{x}{\sqrt{x^2 + 1}} \):
\[
\frac{d}{dx} \left( -\frac{x}{\sqrt{x^2 + 1}} \right) = - \frac{\sqrt{x^2 + 1} \cdot 1 - x \cdot \frac{x}{\sqrt{x^2 + 1}}}{(\sqrt{x^2 + 1})^2}
\]

Simplify:
\[
= - \frac{\sqrt{x^2 + 1} - \frac{x^2}{\sqrt{x^2 + 1}}}{x^2 + 1}
= - \frac{\frac{x^2 + 1 - x^2}{\sqrt{x^2 + 1}}}{x^2 + 1}
= - \frac{1}{(x^2 + 1)^{3/2}}
\]

### Step 4: Combine results
Since the derivative of the constant \(-2\) is zero, the derivative of \( f(x) \) is:
\[
f'(x) = - \frac{1}{(x^2 + 1)^{3/2}}
\]

Thus, the derivative is:
\[
f'(x) = - \frac{1}{(x^2 + 1)^{3/2}}
\]

Would you like any further clarification or additional details?

Here are five related questions to explore further:
1. How would you find the second derivative of this function?
2. What is the behavior of \( f'(x) \) as \( x \to \infty \)?
3. How can you use the chain rule for more complex functions?
4. How would the derivative change if the square root in the denominator were cubed instead?
5. How does this derivative affect the concavity of \( f(x) \)?

**Tip:** The quotient rule is especially useful when differentiating ratios of functions but can often be combined with simpler rules, such as the power or chain rule, for easier computations.

Find the derivative of f(x) = -2 - (x/√(x^2 + 1))

This page provides a detailed step-by-step solution for finding the derivative of the function f(x) = -2 - (x/√(x^2 + 1)) using the quotient and chain rules. Explore how to differentiate functions involving square roots and fractions. Suitable for students in Grades 11-12.

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