Find the derivative of the following functions.
y = e^-x sin x

Verified step by step guidance

1. Step 1: Identify the function y = e^(-x) * sin(x) as a product of two functions, u(x) = e^(-x) and v(x) = sin(x). This suggests using the product rule for differentiation.
2. Step 2: Recall the product rule for differentiation, which states that if y = u(x) * v(x), then the derivative y' = u'(x) * v(x) + u(x) * v'(x).
3. Step 3: Differentiate u(x) = e^(-x). The derivative of e^(-x) with respect to x is -e^(-x).
4. Step 4: Differentiate v(x) = sin(x). The derivative of sin(x) with respect to x is cos(x).
5. Step 5: Apply the product rule: y' = (-e^(-x)) * sin(x) + e^(-x) * cos(x). Combine these results to express the derivative of the original function.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It represents the slope of the tangent line to the curve of the function at any given point. In calculus, derivatives are fundamental for understanding rates of change and are used extensively in optimization problems.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are products of simpler functions, such as the given function y = e^(-x) sin(x).

Exponential and Trigonometric Functions

Exponential functions, like e^(-x), and trigonometric functions, such as sin(x), have specific derivatives that are crucial for differentiation. The derivative of e^x is e^x, and the derivative of sin(x) is cos(x). Understanding these derivatives is vital for applying the product rule effectively in the context of the given function.