Question: Use the chain rule to find the derivative of
10sqrt(4x^3 + 3x^9)
Type your answer without fractional or negative exponents. Use sqrt(x) for \sqrt{x}.
1. **State the problem:** Find the derivative of the function $$f(x) = 10\sqrt{4x^3 + 3x^9}$$ using the chain rule.
2. **Recall the chain rule:** If $$f(x) = g(h(x))$$, then $$f'(x) = g'(h(x)) \cdot h'(x)$$.
3. **Identify outer and inner functions:**
- Outer function: $$g(u) = 10\sqrt{u} = 10u^{1/2}$$
- Inner function: $$h(x) = 4x^3 + 3x^9$$
4. **Differentiate the outer function:**
$$g'(u) = 10 \cdot \frac{1}{2} u^{-1/2} = 5 u^{-1/2} = \frac{5}{\sqrt{u}}$$
5. **Differentiate the inner function:**
$$h'(x) = \frac{d}{dx}(4x^3) + \frac{d}{dx}(3x^9) = 12x^2 + 27x^8$$
6. **Apply the chain rule:**
$$f'(x) = g'(h(x)) \cdot h'(x) = \frac{5}{\sqrt{4x^3 + 3x^9}} \cdot (12x^2 + 27x^8)$$
7. **Simplify the expression:**
$$f'(x) = \frac{5(12x^2 + 27x^8)}{\sqrt{4x^3 + 3x^9}} = \frac{60x^2 + 135x^8}{\sqrt{4x^3 + 3x^9}}$$
8. **Final answer:**
$$\boxed{f'(x) = \frac{60x^2 + 135x^8}{\sqrt{4x^3 + 3x^9}}}$$