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

Verified step by step guidance

1. 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'.
2. 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}.
3. 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.
4. 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}.
5. 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.

Key Concepts

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.