Use the chain rule to find the derivative of
f
(
x
)
=

6
√
3
x
4
+
8
x
10

Type your answer without fractional or negative exponents. Use sqrt(x) for
√
x
.
f
'
(
x
)
=

To find the derivative of the function \( f(x) = 6 \sqrt{3x^4 + 8x^{10}} \), we will use the chain rule.

### Given:
\[ f(x) = 6 \sqrt{3x^4 + 8x^{10}} \]

### Step-by-step solution:

1. **Rewrite the function for differentiation**:
   \[
   f(x) = 6 (3x^4 + 8x^{10})^{1/2}
   \]

2. **Apply the chain rule**:
   The chain rule states that if \( h(x) = (g(x))^n \), then \( h'(x) = n(g(x))^{n-1} \cdot g'(x) \).
   
   Here, \( g(x) = 3x^4 + 8x^{10} \) and \( n = \frac{1}{2} \).

3. **Differentiate using the chain rule**:
   \[
   f'(x) = 6 \cdot \frac{1}{2} (3x^4 + 8x^{10})^{-1/2} \cdot (12x^3 + 80x^9)
   \]

4. **Simplify the expression**:
   - Multiply the constants:
     \[
     f'(x) = 3 (3x^4 + 8x^{10})^{-1/2} \cdot (12x^3 + 80x^9)
     \]
   - Rewrite the expression without negative exponents:
     \[
     f'(x) = \frac{3 (12x^3 + 80x^9)}{\sqrt{3x^4 + 8x^{10}}}
     \]

### Final Answer:
\[
f'(x) = \frac{3(12x^3 + 80x^9)}{\sqrt{3x^4 + 8x^{10}}}
\]

Would you like a detailed explanation of any particular step? Here are some related questions you might find useful:

1. How does the chain rule work for functions with roots?
2. What is the derivative of \( \sqrt{x} \) using the chain rule?
3. How do you simplify derivatives involving roots and powers?
4. What is the derivative of \( x^n \) for any integer \( n \)?
5. How do you apply the product rule when differentiating?

**Tip:** Always simplify the inner function first when using the chain rule; it makes differentiation easier.

Use the chain rule to find the derivative of f(x) = 6√(3x^4 + 8x^10). Type your answer without fractional or negative exponents. Use sqrt(x) for √x. f'(x) =

Learn how to apply the chain rule to differentiate the function f(x) = 6√(3x^4 + 8x^10). This calculus problem involves using the chain rule, differentiation of polynomials, and simplifying the expression. Suitable for students in Grades 11-12 or Calculus I.

Math Problem Statement

Use the chain rule to find the derivative of f ( x )

Type your answer without fractional or negative exponents. Use sqrt(x) for √ x . f ' ( x )

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