Evaluate and simplify y'.
y = (3x+5)¹⁰ √x²+5 / (x³+1)⁵⁰

검증된 단계별 안내

First, identify the function y given by y = (3x+5)^{10} \(\sqrt{x^2+5}\) / (x^3+1)^{50}. We need to find the derivative y'.

Apply the product rule for differentiation. The product rule states that if you have a function h(x) = u(x) * v(x), then h'(x) = u'(x) * v(x) + u(x) * v'(x). Here, consider u(x) = (3x+5)^{10} and v(x) = \(\sqrt{x^2+5}\) / (x^3+1)^{50}.

Differentiate u(x) = (3x+5)^{10} using the chain rule. The chain rule states that if you have a composite function f(g(x)), then the derivative is f'(g(x)) * g'(x). Here, f(x) = x^{10} and g(x) = 3x+5.

Differentiate v(x) = \(\sqrt{x^2+5}\) / (x^3+1)^{50} using the quotient rule. The quotient rule states that if you have a function q(x) = n(x) / d(x), then q'(x) = (n'(x) * d(x) - n(x) * d'(x)) / (d(x))^2. Here, n(x) = \(\sqrt{x^2+5}\) and d(x) = (x^3+1)^{50}.

Combine the derivatives obtained from the product rule, chain rule, and quotient rule to find y'. Simplify the expression by combining like terms and reducing any fractions if possible.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this context, we need to apply differentiation rules to the given function y, which is a combination of polynomial and radical expressions. Understanding how to differentiate these types of functions is essential for evaluating y'.

Product Rule

The Product Rule is a formula used to differentiate products of two or more functions. It states that if you have two functions u(x) and v(x), the derivative of their product is given by u'v + uv'. In the given function y, we will likely need to apply the Product Rule since it is a quotient of two functions, which can be treated as a product when applying the Quotient Rule.

Quotient Rule

The Quotient Rule is used to differentiate a function that is the ratio of two other functions. It states that if y = u/v, then y' = (u'v - uv')/v². This rule is crucial for simplifying the derivative of the function y provided in the question, as it allows us to handle the division of the two components effectively.

Evaluate and simplify y'.y = (3x+5)¹⁰ √x²+5 / (x³+1)⁵⁰