Find f′(x), f′′(x), and f′′′(x) for the following functions.
f(x) = 3x³ + 5x² + 6x

Verified step by step guidance

1. To find the first derivative, f′(x), apply the power rule to each term of the function f(x) = 3x^3 + 5x^2 + 6x. The power rule states that d/dx [x^n] = n*x^(n-1).
2. For the term 3x^3, apply the power rule: the derivative is 3 * 3 * x^(3-1) = 9x^2.
3. For the term 5x^2, apply the power rule: the derivative is 5 * 2 * x^(2-1) = 10x.
4. For the term 6x, apply the power rule: the derivative is 6 * 1 * x^(1-1) = 6.
5. Combine the derivatives of each term to get f′(x) = 9x^2 + 10x + 6. To find f′′(x) and f′′′(x), repeat the differentiation process on f′(x) using the power rule.

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. The first derivative, f'(x), indicates the slope of the tangent line to the curve at any point, while higher-order derivatives, such as f''(x) and f'''(x), provide information about the curvature and acceleration of the function.

Power Rule

The Power Rule is a fundamental technique in calculus used to differentiate functions of the form f(x) = ax^n, where a is a constant and n is a real number. According to this rule, the derivative f'(x) is calculated as f'(x) = n * ax^(n-1). This rule simplifies the differentiation process, especially for polynomial functions like the one given in the question.

Higher-Order Derivatives

Higher-order derivatives are derivatives of derivatives, providing deeper insights into the behavior of a function. The second derivative, f''(x), indicates the concavity of the function, while the third derivative, f'''(x), can reveal information about the rate of change of the curvature. Understanding these derivatives is essential for analyzing the function's behavior beyond just its slope.