Answer :
Final answer:
By applying the chain rule to find the derivative of 4√(5x^{10}+4x^{4}), the process involves differentiating the outer function with respect to the inner function and then multiplying by the derivative of the inner function. The correct answer matches option A: 4(5x^{10}+4x^{4})^{−1/2}(50x^{9}+16x^{3}).
Explanation:
To find the derivative of 4√(5x^{10}+4x^{4}) using the chain rule, we identify the outer function as the square root (or √(x)) and the inner function as (5x^{10}+4x^{4}). Applying the chain rule, we first differentiate the outer function with respect to the inner function, then multiply by the derivative of the inner function with respect to x.
Derivative of the outer function: √(g(x))' = √g'(x) = (g(x))^{−1/2}
Derivative of the inner function: (5x^{10}+4x^{4})' = 50x^{9}+16x^{3}
Multiplying the derivatives together and applying the constant 4 from the original function, we get:
4(5x^{10}+4x^{4})^{−1/2}(50x^{9}+16x^{3}).
This matches option A exactly, suggesting that option A is the correct answer to the question.
