Given that f(x) = 20√(x^2+36) what whole number is equal to f'(8)

To find the derivative \( f'(x) \) of the function \( f(x) = 20\sqrt{x^2 + 36} \), and then evaluate it at \( x = 8 \), follow these steps:

1. **Rewrite the function for easier differentiation:**

   \[
   f(x) = 20(x^2 + 36)^{1/2}
   \]

2. **Differentiate using the chain rule:**

   The chain rule states that if \( y = g(h(x)) \), then \( y' = g'(h(x)) \cdot h'(x) \).
   
   Let \( u(x) = x^2 + 36 \), so \( f(x) = 20u(x)^{1/2} \).

   First, differentiate the outer function with respect to \( u \):

   \[
   \frac{d}{du} \left( 20u^{1/2} \right) = 20 \cdot \frac{1}{2}u^{-1/2} = 10u^{-1/2}
   \]

   Now, multiply by the derivative of the inner function \( u(x) = x^2 + 36 \):

   \[
   \frac{du}{dx} = 2x
   \]

   So the derivative \( f'(x) \) is:

   \[
   f'(x) = 10(x^2 + 36)^{-1/2} \cdot 2x = \frac{20x}{\sqrt{x^2 + 36}}
   \]

3. **Evaluate \( f'(x) \) at \( x = 8 \):**

   \[
   f'(8) = \frac{20 \cdot 8}{\sqrt{8^2 + 36}} = \frac{160}{\sqrt{64 + 36}} = \frac{160}{\sqrt{100}} = \frac{160}{10} = 16
   \]

The whole number equal to \( f'(8) \) is \( \boxed{16} \).

Would you like further explanation or have any other questions?

Here are some related questions:
1. How do you apply the chain rule in more complex functions?
2. What are the common mistakes to avoid when differentiating square roots?
3. How does the derivative of a function involving square roots change when evaluating at different points?
4. Can you interpret the physical meaning of \( f'(x) \) in a real-world context?
5. How would the derivative change if the function were \( f(x) = 20\sqrt{x^2 + k} \) with a different constant \( k \)?

**Tip:** When differentiating a function involving square roots, always simplify the derivative as much as possible before substituting the specific value of \( x \).

Learn how to find the derivative f'(x) of the function f(x) = 20√(x^2 + 36) using the chain rule and evaluate it at x = 8. This problem covers differentiation, chain rule applications, and square root function derivatives. Suitable for advanced high school or college-level students.

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