Derivatives Find and simplify the derivative of the following functions.
f(x) = 3x⁴(2x²−1)

Verified step by step guidance

1. Identify the function as a product of two functions: f(x) = u(x) * v(x), where u(x) = 3x⁴ and v(x) = (2x²−1).
2. Apply the product rule for derivatives, which states that the derivative of a product u(x)v(x) is u'(x)v(x) + u(x)v'(x).
3. Find the derivative of u(x) = 3x⁴. The derivative, u'(x), is obtained by applying the power rule: u'(x) = 12x³.
4. Find the derivative of v(x) = 2x²−1. The derivative, v'(x), is obtained by applying the power rule: v'(x) = 4x.
5. Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula: f'(x) = 12x³(2x²−1) + 3x⁴(4x). Simplify the expression to find the derivative of f(x).

Key Concepts

Derivatives

A derivative represents the rate at which a function changes at a given point. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative can be interpreted as the slope of the tangent line to the graph of the function at a specific point.

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 multiplied together, as in the case of f(x) = 3x⁴(2x²−1).

Simplification of Derivatives

After finding the derivative of a function, simplification is often necessary to express the result in a more manageable form. This may involve combining like terms, factoring, or reducing fractions. Simplifying the derivative helps in understanding the behavior of the function and makes it easier to analyze critical points and concavity.