Find the derivative of 3/(x^5 + 2x^2 - 3) using chain rule

To find the derivative of \( f(x) = \frac{3}{x^5 + 2x^2 - 3} \) using the chain rule, we can rewrite it in a more convenient form and then apply the necessary differentiation rules. 

### Step 1: Rewrite the function
The given function can be rewritten as:
\[
f(x) = 3 \cdot (x^5 + 2x^2 - 3)^{-1}
\]
This allows us to easily apply the chain rule.

### Step 2: Differentiate using the chain rule
The chain rule states that if you have a composite function \( f(g(x)) \), then:
\[
\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
\]
In this case:
- The outer function is \( f(g(x)) = 3 \cdot u^{-1} \), where \( u = x^5 + 2x^2 - 3 \).
- The inner function is \( g(x) = x^5 + 2x^2 - 3 \).

#### Step 3: Derivative of the outer function
For \( f(u) = 3 \cdot u^{-1} \), the derivative is:
\[
f'(u) = -3 \cdot u^{-2}
\]
So, applying this to \( u = x^5 + 2x^2 - 3 \), we get:
\[
f'(g(x)) = -3 \cdot (x^5 + 2x^2 - 3)^{-2}
\]

#### Step 4: Derivative of the inner function
The derivative of the inner function \( g(x) = x^5 + 2x^2 - 3 \) is:
\[
g'(x) = 5x^4 + 4x
\]

### Step 5: Apply the chain rule
Now, multiply the derivative of the outer function by the derivative of the inner function:
\[
f'(x) = -3 \cdot (x^5 + 2x^2 - 3)^{-2} \cdot (5x^4 + 4x)
\]

Thus, the derivative of \( \frac{3}{x^5 + 2x^2 - 3} \) is:
\[
f'(x) = \frac{-3(5x^4 + 4x)}{(x^5 + 2x^2 - 3)^2}
\]

Would you like a deeper explanation of any part? 

Here are five related questions:
1. How is the product rule different from the chain rule?
2. What is the derivative of \( \frac{1}{x^n} \) for any \( n \)?
3. How would you differentiate \( \frac{3}{x^2 + 4x + 5} \)?
4. How do you handle higher-order derivatives using the chain rule?
5. Can implicit differentiation be combined with the chain rule?

**Tip**: Always remember to simplify the inner function before applying the chain rule to avoid mistakes.

Learn how to differentiate the function 3/(x^5 + 2x^2 - 3) using the chain rule. This step-by-step solution covers key concepts such as the chain rule and power rule, suitable for college or AP Calculus students.

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