Find the derivatives of the functions in Exercises 1–42.


𝔂 = (x + 1)² (x² + 2x)

Verified step by step guidance

1. Identify the function as a product of two functions: \( u(x) = (x + 1)^2 \) and \( v(x) = (x^2 + 2x) \).
2. Apply the product rule for differentiation, which states that if \( y = u(x) \cdot v(x) \), then \( y' = u'(x) \cdot v(x) + u(x) \cdot v'(x) \).
3. Differentiate \( u(x) = (x + 1)^2 \) using the chain rule: \( u'(x) = 2(x + 1) \cdot 1 = 2(x + 1) \).
4. Differentiate \( v(x) = x^2 + 2x \) using the power rule: \( v'(x) = 2x + 2 \).
5. Substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into the product rule formula to find \( y' \).

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 denoted as f'(x) or dy/dx.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, 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.

Chain Rule

The chain rule is a method for differentiating composite functions. If a function y is defined as a composition of two functions, such as y = f(g(x)), the chain rule states that the derivative is dy/dx = f'(g(x)) * g'(x). This rule is crucial when dealing with functions that are nested within each other, which may occur in more complex expressions.