Calculate the derivative of the following functions.
y = (p+3)² sin p²

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative is often denoted as f'(x) or dy/dx, and it can be calculated using various rules, such as the power rule, product rule, and chain rule.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. If u(p) and v(p) are two differentiable functions, the product rule states that the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are multiplied together, as seen in the given function y = (p+3)² sin p².

Chain Rule

The chain rule is a technique for differentiating composite functions, which are functions within functions. If a function y = f(g(p)) is composed of an outer function f and an inner function g, the chain rule states that the derivative is f'(g(p)) * g'(p). This rule is particularly useful when dealing with functions that involve powers or trigonometric functions, as in the case of sin p² in the given problem.