Evaluate and simplify y'.
y = (x²+1)³ / (x⁴+7)⁸(2x+1)⁷

Verified step by step guidance

1. First, identify the function y in the equation y'.y = (x²+1)³ / (x⁴+7)⁸(2x+1)⁷. Here, y is implicitly defined, and y' represents the derivative of y with respect to x.
2. To find y', use implicit differentiation. Differentiate both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating y, apply the chain rule: d(y)/dx = y'.
3. Differentiate the right-hand side of the equation using the quotient rule. The quotient rule states that if you have a function f(x)/g(x), its derivative is (f'(x)g(x) - f(x)g'(x)) / (g(x))².
4. Apply the chain rule to differentiate the terms (x²+1)³, (x⁴+7)⁸, and (2x+1)⁷. For example, the derivative of (x²+1)³ is 3(x²+1)² * 2x, using the chain rule.
5. After differentiating, simplify the expression by combining like terms and factoring where possible. This will give you the expression for y' in terms of x.

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 quotient of two complex expressions. Understanding how to differentiate polynomial and composite functions is essential for solving the problem.

Quotient Rule

The Quotient Rule is a formula used to differentiate functions that are expressed as the ratio of two other functions. It states that if y = u/v, then y' = (u'v - uv')/v², where u and v are functions of x. This rule is crucial for this problem since y is a quotient, and applying it correctly will allow us to find y'.

Chain Rule

The Chain Rule is a fundamental technique in calculus for differentiating composite functions. It states that if a function y is composed of another function u, then the derivative y' can be found by multiplying the derivative of u with respect to x by the derivative of y with respect to u. This concept is important here as the function y involves powers of polynomials, requiring the Chain Rule for proper differentiation.