Question

Use the definition of the derivative to compute the derivative of the given function. $$f(x)=3 x^{2}-x+4$$

Short answer

The derivative of the function is \( f'(x) = 6x - 1 \).

Step-by-step solution

Understand the Derivative Definition

The derivative of a function \( f(x) \) at a point \( x \) is defined as the limit: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] We will apply this definition to find the derivative of \( f(x) = 3x^2 - x + 4 \).

Substitute in the Function

First, substitute \( x+h \) into the function \( f \): \[ f(x+h) = 3(x+h)^2 - (x+h) + 4 \] Simplify this expression.

Expand \( f(x+h) \)

Expand the quadratic expression: \[ 3(x+h)^2 = 3(x^2 + 2xh + h^2) = 3x^2 + 6xh + 3h^2 \] Thus, \[ f(x+h) = 3x^2 + 6xh + 3h^2 - x - h + 4 \]

Form the Difference Quotient

Form the difference quotient: \[ \frac{f(x+h) - f(x)}{h} = \frac{(3x^2 + 6xh + 3h^2 - x - h + 4) - (3x^2 - x + 4)}{h} \] Simplify the numerator.

Simplify the Numerator

Simplify the numerator: \[ (3x^2 + 6xh + 3h^2 - x - h + 4) - (3x^2 - x + 4) = 6xh + 3h^2 - h \] The 3x^2, -x, and 4 terms cancel out.

Simplify the Fraction

Simplify the expression by factoring \( h \) from the numerator: \[ \frac{6xh + 3h^2 - h}{h} = \frac{h(6x + 3h - 1)}{h} \] Cancel out the \( h \) in the numerator and denominator: \[ 6x + 3h - 1 \]

Calculate the Limit

Calculate the limit as \( h \) approaches 0: \[ \lim_{h \to 0} (6x + 3h - 1) = 6x - 1 \] Therefore, the derivative \( f'(x) = 6x - 1 \).

Key concepts

Limit Definition of Derivative

Understanding the limit definition of a derivative is crucial in calculus as it lays the foundation for computing derivatives from first principles. The derivative of a function, denoted as \( f'(x) \), represents the slope of the tangent line to the curve at any point \( x \). This concept is important because it tells us how the function behaves or changes at that point.

The limit definition of the derivative is given by the formula:

• \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

This formula calculates the derivative by taking the average rate of change between the points \( x \) and \( x+h \), and then finding what this average rate approaches as \( h \) becomes infinitely small. Hence, the derivative can be thought of as an instantaneous rate of change.

For example, in the case of the polynomial function \( f(x) = 3x^2 - x + 4 \), applying this definition allows us to find its derivative, giving insight into its rate of change.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number exponents, combined using addition, subtraction, and multiplication. They are one of the simplest types of functions in calculus and algebra.

A polynomial in one variable \( x \) of degree \( n \) can be written in the form:

• \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)

Where \( a_n, a_{n-1}, \ldots, a_0 \) are constants, and \( a_n eq 0 \). Each term represents a power of \( x \) with a coefficient.

In the given function \( f(x) = 3x^2 - x + 4 \), we have a second-degree polynomial, also known as a quadratic function. It's composed of three terms:

• \( 3x^2 \), where \( 3 \) is the coefficient of the term.
• \(-x\), which is equivalent to \(-1x\).
• \(+4\), a constant term.

Polynomial functions are smooth and continuous, making them easier to work with, which is why they are often used to model real-world situations in disciplines like physics and economics.

Calculus Problem Solving

Solving calculus problems often involves applying fundamental theorems and formulas, understanding functions, and breaking down complex tasks into simpler parts. Successful calculus problem solving relies on several techniques and strategies.

One key technique is breaking problems into steps: identifying the kind of problem, choosing the right methods and tools, and then following a logical sequence. The original exercise illustrates this with step-by-step instructions:

1. **Recognize a Problem Type:** Determine if using the limit definition of a derivative is appropriate for solving the given problem.
2. **Apply the Definition:** Substitute the function into the formula and follow algebraic rules to simplify the expression.
3. **Simplify Expressions:** Use algebra to reduce complexity, like factoring terms and canceling out common factors.

Each step builds on the previous one, leading to a clear solution. In the exercise, applying these principles efficiently derived the derivative \( f'(x) = 6x - 1 \) from the polynomial \( f(x) = 3x^2 - x + 4 \). Thus, breaking down complex problems helps organize thoughts and manage calculations, ultimately making calculus more approachable.