Evaluate and simplify y'.
y = 4u²+u / 8u+1

Verified step by step guidance

1. First, identify the expression given: \( y' \cdot y = \frac{4u^2 + u}{8u + 1} \). This is a differential equation involving \( y' \), the derivative of \( y \) with respect to \( u \).
2. To simplify the expression, consider factoring the numerator \( 4u^2 + u \). Notice that \( u \) is a common factor, so factor it out: \( u(4u + 1) \).
3. Next, check if the expression can be simplified further by canceling common factors in the numerator and denominator. The denominator is \( 8u + 1 \), which does not share any common factors with the numerator after factoring.
4. Now, consider the context of the problem. If \( y' \cdot y \) represents a derivative, you might need to separate variables or integrate to solve for \( y \). However, without additional context, focus on simplifying the algebraic expression.
5. Finally, ensure the expression is in its simplest form. Since no further simplification is possible, the expression remains \( \frac{u(4u + 1)}{8u + 1} \).

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the curve at any given point. In this context, we need to apply the rules of differentiation to find y', the derivative of the function y with respect to u.

Quotient Rule

The quotient rule is a formula used to differentiate functions that are expressed as the ratio of two other functions. If y = f(u)/g(u), the derivative y' is given by (f' * g - f * g') / g². This rule is essential for the given problem since y is defined as a fraction, and applying the quotient rule will allow us to find the derivative correctly.

Simplification

Simplification in calculus involves reducing an expression to its most basic form, making it easier to interpret or compute. After finding the derivative using the quotient rule, we may need to combine like terms, factor, or cancel common factors to present the result in a clearer and more concise manner. This step is crucial for ensuring the final answer is both accurate and easy to understand.