7–28. Derivatives Evaluate the following derivatives.


d/dy (y^{sin y})

Verified step by step guidance

1. Step 1: Recognize that the function y^{sin(y)} is a composite function involving both a base (y) and an exponent (sin(y)). To differentiate it, use the logarithmic differentiation method.
2. Step 2: Take the natural logarithm of both sides to simplify the differentiation process. Let f(y) = y^{sin(y)}, then ln(f(y)) = sin(y) * ln(y).
3. Step 3: Differentiate both sides with respect to y. Use the chain rule on the left-hand side and the product rule on the right-hand side. The derivative of ln(f(y)) is (1/f(y)) * f'(y), and the derivative of sin(y) * ln(y) is [cos(y) * ln(y)] + [sin(y) * (1/y)].
4. Step 4: Solve for f'(y) by multiplying through by f(y). Recall that f(y) = y^{sin(y)}, so f'(y) = y^{sin(y)} * {[cos(y) * ln(y)] + [sin(y) * (1/y)]}.
5. Step 5: Write the final expression for the derivative. The derivative of y^{sin(y)} with respect to y is y^{sin(y)} * {[cos(y) * ln(y)] + [sin(y) * (1/y)]}.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that allows us to understand how a function behaves as its input changes. The notation d/dy indicates that we are taking the derivative with respect to the variable y.

Chain Rule

The chain rule is a formula for computing the derivative of a composite function. It states that if you have a function that is composed of two or more functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when differentiating functions like y^{sin y}, where both y and sin y are involved.

Exponential Functions

Exponential functions are functions of the form f(x) = a^g(x), where a is a constant and g(x) is a function of x. In the context of the given problem, y^{sin y} can be rewritten using properties of exponents, which can simplify the differentiation process. Understanding how to manipulate and differentiate exponential functions is crucial for solving the derivative in this question.