Question

Differentiate from first principles \(f(x)=2 x^{3}\)

Short answer

The derivative, using first principles, is \(f'(x) = 6x^2\).

Step-by-step solution

Understand the Derivative from First Principles

The derivative of a function, from first principles, can be calculated using the limit definition. The formula is:\[f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}\]We will apply this definition to the function \(f(x) = 2x^3\).

Substitute into the Limit Definition

First, calculate \(f(x + h)\):\[f(x + h) = 2(x + h)^3\]Expanding \((x + h)^3\), we have:\[(x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\]Therefore:\[f(x + h) = 2(x^3 + 3x^2h + 3xh^2 + h^3) = 2x^3 + 6x^2h + 6xh^2 + 2h^3\]

Evaluate the Expression in the Difference

Now, substitute \(f(x + h)\) and \(f(x)\) into the limit expression:\[f'(x) = \lim_{{h \to 0}} \frac{{2x^3 + 6x^2h + 6xh^2 + 2h^3 - (2x^3)}}{h}\]Simplify the numerator:\[= \lim_{{h \to 0}} \frac{{6x^2h + 6xh^2 + 2h^3}}{h}\]

Simplify and Cancel Terms

Factor \(h\) from the numerator:\[= \lim_{{h \to 0}} \frac{h(6x^2 + 6xh + 2h^2)}{h}\]Cancel \(h\) from the numerator and denominator:\[= \lim_{{h \to 0}} (6x^2 + 6xh + 2h^2)\]

Evaluate the Limit

Finally, evaluate the limit as \(h\) approaches 0:\[f'(x) = 6x^2 + 6x(0) + 2(0)^2 = 6x^2\]This means the derivative of \(f(x) = 2x^3\) is \(f'(x) = 6x^2\).

Key concepts

Limit Definition of Derivative

The limit definition of the derivative is a foundational concept in calculus. It provides a mathematical way of finding the rate of change of a function at a specific point. When we talk about derivatives, we essentially mean the slope of the tangent line to the curve of a function at a certain point.
To understand this, think about how a car's speed is the rate of change of its position over time. A derivative tells us how fast a function's value changes as its input changes.

• The standard formula for the derivative using this definition is: \[f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}\]
• Here, \(h\) represents a very small change in \(x\), and \(f(x+h) - f(x)\) represents the change in the function's value.
• The limit as \(h\) approaches zero gives us the instantaneous rate of change, or the derivative.

By substituting \(f(x) = 2x^3\) into this formula, we can work through the steps to find its derivative from first principles.

Polynomial Differentiation

Polynomial differentiation is a process used to find the derivative of a polynomial function. It involves applying differentiation rules, often starting with first principles, as demonstrated earlier. When you're working with polynomials, each term needs to be differentiated separately.
Consider the polynomial function \(f(x) = 2x^3\). The process involves a few important steps:

• Expansion and Simplification: If you have a complicated expression, like \((x+h)^3\), expand it first. This gives terms that can be differentiated individually.
• Applying the Limit: Substitute these expanded terms into the limit definition and simplify.
• Cancel Terms: Factors like \(h\) simplify the expression and are cleared out by canceling accordingly.

This makes polynomial differentiation systematic, especially for higher-degree terms. Each term's exponent gets lessened by one, revealing the derivative's structure.

Basic Calculus Concepts

Understanding basic calculus concepts is vital for tackling differentiation and integration problems effectively. These concepts provide the tools and language used throughout calculus.

• Functions and Their Graphs: Functions like \(f(x) = 2x^3\) translate into curves or lines on a graph, helping visualize their behavior.
• Instantaneous Rate of Change: Derivatives give us insight into how a function changes at any given point, akin to a snapshot of motion in physics.
• Limits: The idea of making values "infinitely small" or "infinitely large" is central to calculus, as it allows us to calculate precise rates of change.

By mastering these fundamental concepts, solving calculus problems, such as differentiating a polynomial function from first principles, becomes clearer and more intuitive.