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.