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Magazine Chapter 23: Problem 16
In Exercises \(3-26,\) find the derivative of each of the functions by using the definition. $$y=2 x-4 x^{3}$$
Short Answer
Expert verified
The derivative is \(f'(x) = 2 - 12x^2\).
Step by step solution
01
Understand the Problem
We need to find the derivative of the function \(y = 2x - 4x^3\) using the definition of the derivative. This means we'll apply the limit process to calculate it.
02
Use the Definition of the Derivative
The derivative of a function \(f(x)\) is given by the definition \(f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}\). We will apply this definition to our function \(y = 2x - 4x^3\).
03
Substitute \(f(x)\) and \(f(x+h)\) into the Definition
Calculate \(f(x+h)\):\[f(x+h) = 2(x+h) - 4(x+h)^3\].First, expand \((x+h)^3\) using the binomial theorem:\[(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\].Thus,\[f(x+h) = 2x + 2h - 4(x^3 + 3x^2h + 3xh^2 + h^3)\].
04
Simplify \(f(x+h) - f(x)\)
Subtract \(f(x)\) from \(f(x+h)\):\[f(x+h) - f(x) = (2x + 2h - 4x^3 - 12x^2h - 12xh^2 - 4h^3) - (2x - 4x^3)\].Cancel the common terms:\[= 2h - 12x^2h - 12xh^2 - 4h^3\].
05
Factor and Simplify
Factor \(h\) out from the expression:\[= h(2 - 12x^2 - 12xh - 4h^2)\].
06
Take the Limit as \(h \to 0\)
Evaluate the limit:\[f'(x) = \lim_{{h \to 0}} \frac{{h(2 - 12x^2 - 12xh - 4h^2)}}{h}\].This simplifies to:\[f'(x) = \lim_{{h \to 0}} (2 - 12x^2 - 12xh - 4h^2)\].
07
Apply the Limit
Substitute \(h = 0\):\[f'(x) = 2 - 12x^2\].This is the derivative of the function.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Derivative
The derivative is a fundamental concept in calculus that represents the rate at which a function is changing at any given point. In simpler terms, it's like finding the slope of the curve of a function at a particular point on its graph. The symbolic representation of a derivative is usually given as \( f'(x) \) or \( \frac{dy}{dx} \). The operation of deriving is called differentiation.
- The derivative tells us how steep the graph is.
- It is calculated using a formula that involves limits.
- In physics and engineering, derivatives can represent speeds, accelerations, and other rates of change.
Limit Process
The limit process is an essential mathematical technique used to calculate the derivative of a function. It involves evaluating a function as it approaches a specific point. The idea is to understand how a function behaves as the input gets very close to a particular value.
- In the definition of the derivative, the limit is taken as \( h \) approaches 0.
- Expressed, it looks like this: \( f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \).
- This process helps us find the exact derivative value rather than simply estimating it.
Binomial Theorem
The binomial theorem provides a way to expand expressions that are raised to a power, such as \((x+h)^3\) in our example. It's a handy tool that makes it easier to handle polynomials.
- The theorem states: \((x + h)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} h^k\).
- This formula lets us expand the powers of binomials efficiently.
- In the context of differentiation, expanding terms like \((x+h)^3\) helps us simplify expressions to apply the limit process.
Function Differentiation
Function differentiation refers to the process of finding the derivative of a function. Using the definition of a derivative and applying the limit process, a function is differentiated to find its rate of change.
- Start by understanding the function you need to differentiate.
- Substitute and simplify the expressions as seen in the original exercise.
- Apply limit operations carefully to reach the final derivative.
