Find the derivative of the following functions.
y = x sin x

Verified step by step guidance

1. Identify the function as a product of two functions: \( y = x \cdot \sin(x) \). Here, \( u = x \) and \( v = \sin(x) \).
2. Apply the product rule for differentiation, which states that if \( y = u \cdot v \), then \( \frac{dy}{dx} = u'v + uv' \).
3. Differentiate \( u = x \) with respect to \( x \). The derivative \( u' = \frac{d}{dx}(x) = 1 \).
4. Differentiate \( v = \sin(x) \) with respect to \( x \). The derivative \( v' = \frac{d}{dx}(\sin(x)) = \cos(x) \).
5. Substitute \( u' = 1 \), \( v = \sin(x) \), \( u = x \), and \( v' = \cos(x) \) into the product rule formula: \( \frac{dy}{dx} = 1 \cdot \sin(x) + x \cdot \cos(x) \).

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. Derivatives are fundamental in calculus for understanding rates of change and are used in various applications, including optimization and motion analysis.

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', where u' and v' are the derivatives of u and v, respectively. This rule is essential when differentiating functions that are multiplied together, such as in the given function y = x sin x.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental functions in mathematics that relate angles to ratios of sides in right triangles. The sine function, sin(x), is particularly important in calculus as it has well-defined derivatives and integrals. Understanding the properties and derivatives of these functions is crucial when working with problems involving trigonometric expressions, like in the function y = x sin x.