In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
124. x^(sin y) = ln y

Verified step by step guidance

1. Start with the given equation: \(x^{\sin y} = \ln y\).
2. Since the equation involves both \(x\) and \(y\) in a complicated way, use implicit differentiation with respect to \(x\). Remember that \(y\) is a function of \(x\), so when differentiating terms involving \(y\), apply the chain rule.
3. Differentiate the left side \(x^{\sin y}\) using logarithmic differentiation: first take the natural logarithm of both sides to simplify the exponentiation. That is, write \(\ln(x^{\sin y}) = \ln(\ln y)\).
4. Simplify the left side using logarithm properties: \(\sin y \cdot \ln x = \ln(\ln y)\).
5. Now differentiate both sides with respect to \(x\). For the left side, use the product rule on \(\sin y \cdot \ln x\), remembering that \(y\) depends on \(x\). For the right side, use the chain rule on \(\ln(\ln y)\).

Key Concepts

Implicit Differentiation

Implicit differentiation is used when a function is defined implicitly rather than explicitly. It involves differentiating both sides of an equation with respect to the independent variable, treating dependent variables as functions of that variable, and then solving for the derivative.

Logarithmic Differentiation

Logarithmic differentiation simplifies differentiation of functions where variables appear as exponents or products by taking the natural logarithm of both sides. This transforms complicated expressions into sums or products, making differentiation more manageable.

Chain Rule

The chain rule is a fundamental differentiation technique used when differentiating composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.