Final answer:
The derivative of the function 10 √(4 x⁹ + 7 x¹⁰) using the chain rule is 5 · (36x⁸ + 70x⁹) · (4x⁹ + 7x¹⁰)⁻½.
Step-by-step explanation:
To use the chain rule to find the derivative of the function 10 √(4 x⁹ + 7 x¹⁰), let's denote the inner function inside the square root as u(x) = 4 x⁹ + 7 x¹⁰. The outer function is f(u) = 10√u.
The chain rule states that the derivative of a composite function f(g(x)) is (f’(g(x)) · g’(x)). For our function, we can write it in the form f(u(x)) and use the rule accordingly.
First, find the derivative of the outer function with respect to u, which is ¼ u⁻½. When you plug in u(x), you have ¼ (4 x⁹ + 7 x¹⁰)⁻½. Then, you need to multiply this by the derivative of u with respect to x, which is 9· 4x⁸ + 10· 7x⁹.
Derivative steps:
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- Identify the inner function u(x) = 4x⁹ + 7x¹⁰.
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- Compute the derivative of the outer function f(u) with respect to u: f’(u) = 10 · ¼u⁻½ = ¼u⁻½.
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- Compute the derivative of u with respect to x: u’(x) = 36x⁸ + 70x⁹.
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- Apply the chain rule: (f’(u(x)) · u’(x)) = ¼(4x⁹ + 7x¹⁰)⁻½ · (36x⁸ + 70x⁹).
Therefore, the derivative of the function 10 √(4 x⁹ + 7 x¹⁰) is 5 · (36x⁸ + 70x⁹) · (4x⁹ + 7x¹⁰)⁻½.