Calculate the derivative of the function.
f(x) = (2x2 − 3)−1
f ′(x) =

The given function is:

\[
f(x) = (2x^2 - 3)^{-1}
\]

To differentiate this function, we'll apply the chain rule.

### Step 1: Use the Chain Rule
The chain rule states that if you have a composite function \( g(h(x)) \), the derivative is:

\[
\frac{d}{dx} g(h(x)) = g'(h(x)) \cdot h'(x)
\]

Here, \( g(h(x)) = h(x)^{-1} \) and \( h(x) = 2x^2 - 3 \). First, we differentiate the outer function, then the inner function.

#### Step 2: Differentiate the outer function
The outer function is \( h(x)^{-1} \), which is \( (something)^{-1} \). The derivative of \( (something)^{-1} \) is:

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

So,

\[
\frac{d}{dx} \left[ (2x^2 - 3)^{-1} \right] = - (2x^2 - 3)^{-2} \cdot \frac{d}{dx} (2x^2 - 3)
\]

#### Step 3: Differentiate the inner function
The inner function is \( 2x^2 - 3 \), and its derivative is:

\[
\frac{d}{dx} (2x^2 - 3) = 4x
\]

#### Step 4: Combine the results
Now, combine the derivative of the outer function and the inner function:

\[
f'(x) = - (2x^2 - 3)^{-2} \cdot 4x
\]

Thus, the derivative of the function is:

\[
f'(x) = \frac{-4x}{(2x^2 - 3)^2}
\]

### Final answer:
\[
f'(x) = \frac{-4x}{(2x^2 - 3)^2}
\]

Would you like more details on any of the steps or have questions?

Here are 5 related questions to expand your understanding:
1. What is the chain rule and how does it apply to composite functions?
2. How would the derivative change if the exponent were different, e.g., \( f(x) = (2x^2 - 3)^{-n} \)?
3. What is the derivative of \( f(x) = (ax^2 + b)^{-1} \) for general constants \(a\) and \(b\)?
4. How do you apply the quotient rule in differentiation?
5. What are the conditions under which \( f(x) = (2x^2 - 3)^{-1} \) is undefined?

**Tip:** Always check if the inner function of a composite function is differentiable before applying the chain rule.

Calculate the derivative of the function. f(x) = (2x^2 − 3)−1 f ′(x) =

Discover how to calculate the derivative of f(x) = (2x^2 − 3)−1 using the chain rule. This solution involves differentiating composite functions and applying the power rule. Ideal for students in advanced calculus classes, Grades 11-12.

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