Evaluate and simplify y'.


y = ln w / w⁵

Verified step by step guidance

1. First, identify the given expression: y'.y = ln(w) / w⁵. This involves differentiation and simplification.
2. Recognize that y' represents the derivative of y with respect to w. We need to differentiate the expression on the right side with respect to w.
3. Apply the quotient rule for differentiation, which states that if you have a function u/v, its derivative is (u'v - uv') / v². Here, u = ln(w) and v = w⁵.
4. Differentiate u = ln(w) to get u' = 1/w. Differentiate v = w⁵ to get v' = 5w⁴.
5. Substitute these derivatives into the quotient rule formula: y' = [(1/w) * w⁵ - ln(w) * 5w⁴] / (w⁵)². Simplify the expression by combining like terms and reducing powers of w where possible.

Key Concepts

Derivative

The derivative of a function measures how the function's output changes as its input 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 w.

Quotient Rule

The quotient rule is a formula used to differentiate functions that are expressed as the quotient of two other functions. It states that if you have a function y = u/v, where both u and v are differentiable, then the derivative y' is given by (v * u' - u * v') / v². This rule is essential for simplifying the differentiation of y = ln(w) / w⁵.

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It has unique properties, such as the derivative of ln(x) being 1/x. Understanding how to differentiate natural logarithmic functions is crucial for solving the given problem, as it involves ln(w) in the expression for y.