Let $ f(x) = \left(\frac{x+2}{x+4}\right)^4 $ and $ f'(x) = $.

Use the chain rule to find the derivative of $ 4\sqrt{5x^{10} + 4x^4} $. Type your answer without fractional or negative exponents. Use $\sqrt{x}$ for $\sqrt{x}$.

Which of the following options correctly represents the derivative?

A) $ 20x^9 + 16x^3 $

B) $ 20x^9 + 8x^3 $

C) $ 40x^9 + 16x^3 $

D) $ 40x^9 + 8x^3 $

The derivative of 4$\sqrt{5x^{10}+4x^{4}}$, found using the chain rule, is 40x^{9} + 16x^{3}. The solution involves applying the chain rule by differentiating the inner and outer functions and then multiplying their derivatives. The correct option is C) 40x^9 + 16x^3

Explanation:

To find the derivative of 4$\sqrt{5x^{10}+4x^{4}}$, we first understand that this involves a chain rule application. The function can be represented as f(x) = [4(g(x))^{1/2}] where g(x) = 5x^{10} + 4x^{4}. Applying the chain rule, which in essence involves taking the derivative of the outer function and multiplying it by the derivative of the inner function, we start by differentiating the outer function.

If the outer function is u^{1/2}, its derivative is (1/2)u^{-1/2}, where u = 5x^{10} + 4x^{4}. Therefore, the derivative of the outer function (with respect to u) is 2(1/2)(5x^{10} + 4x^{4})^{-1/2}, simplifying to (5x^{10} + 4x^{4})^{-1/2}. Next, we differentiate the inner function, g(x) = 5x^{10} + 4x^{4}, which yields 50x^{9} + 16x^{3}. Combining both using the chain rule gives (5x^{10} + 4x^{4})^{-1/2} \cdot (50x^{9} + 16x^{3}), which after simplification and applying the 4 multiplier gives the final derivative as 40x^{9} + 16x^{3}.