Chapter 33: Problem 6
Differentiate from first principles \(f(x)=2 x^{3}\)
Short Answer
Expert verified
The derivative, using first principles, is \(f'(x) = 6x^2\).
Step by step solution
01
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 \(f(x) = 2x^3\).
02
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:\[f(x + h) = 2(x^3 + 3x^2h + 3xh^2 + h^3) = 2x^3 + 6x^2h + 6xh^2 + 2h^3\]
03
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:\[= \lim_{{h \to 0}} \frac{{6x^2h + 6xh^2 + 2h^3}}{h}\]
04
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:\[= \lim_{{h \to 0}} (6x^2 + 6xh + 2h^2)\]
05
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 \(f'(x) = 6x^2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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:
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.
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.