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Magazine Chapter 1: Problem 17
$$ \text { Differentiate. } $$ $$ y=2 x+(x+2)^{3} $$
Short Answer
Expert verified
\( \frac{dy}{dx} = 2 + 3(x+2)^2 \)
Step by step solution
01
Differentiate the first term
Differentiate the first term, which is a linear function. The term is given as: \[ y_1 = 2x \]Using the differentiation rule for a linear function, we get \[ \frac{dy_1}{dx} = 2 \]
02
Identify the second term
The second term is a composite function. It is given by: \[ y_2 = (x+2)^3 \]We'll need to apply the chain rule to differentiate this.
03
Differentiate the outer function using chain rule
First, differentiate the outer function, which is raised to a power: \[ \frac{d}{du} (u^3) = 3u^2 \]Here, \( u = (x+2) \). Substituting back, we get: \[ 3(x+2)^2 \]
04
Differentiate the inner function
Differentiate the inner function \( u = x + 2 \): \[ \frac{du}{dx} = 1 \]
05
Apply the chain rule
Combine the results from Steps 3 and 4 using the chain rule: \[ \frac{dy_2}{dx} = 3(x+2)^2 \cdot 1 = 3(x+2)^2 \]
06
Sum the derivatives
Add the derivatives of the two terms to get the final answer: \[ \frac{dy}{dx} = \frac{dy_1}{dx} + \frac{dy_2}{dx} \] Thus, \[ \frac{dy}{dx} = 2 + 3(x+2)^2 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chain Rule
The chain rule is a fundamental tool in calculus used to differentiate composite functions. It helps us find the derivative of a function composed of two or more functions. Here's how the chain rule works:
Suppose you have a function \(y\) that is a composite of two functions \(u\) and \(v\), such that \[ y = f(g(x)) \] To differentiate \(y\) with respect to \(x\), we need to follow these steps:
Suppose you have a function \(y\) that is a composite of two functions \(u\) and \(v\), such that \[ y = f(g(x)) \] To differentiate \(y\) with respect to \(x\), we need to follow these steps:
- Differentiate the outer function with respect to the inner function. This means taking the derivative of \(f\) with respect to \(u\).
- Next, differentiate the inner function \(g(x)\) with respect to \(x\).
- Multiply the results of these two differentiations to get the final derivative.
Linear Function
A linear function is one of the simplest forms of functions in mathematics. It is of the form: \[ y = mx + c \] where \(m\) and \(c\) are constants. The slope \(m\) represents how steep the line is, while \(c\) is the y-intercept, showing where the line crosses the y-axis.
When differentiating a linear function, the process is straightforward because the derivative of a linear term \(mx\) is simply the constant \(m\). This is because the rate of change (slope) is constant.
In the provided solution:
\ y_1 = 2x \ Differentiating \(2x\) with respect to \(x\) gives us a constant derivative, \(2\). This intuitive property of linear functions makes their differentiation straightforward.
When differentiating a linear function, the process is straightforward because the derivative of a linear term \(mx\) is simply the constant \(m\). This is because the rate of change (slope) is constant.
In the provided solution:
\ y_1 = 2x \ Differentiating \(2x\) with respect to \(x\) gives us a constant derivative, \(2\). This intuitive property of linear functions makes their differentiation straightforward.
Composite Function
A composite function is a function that is formed by combining two or more functions. Essentially, the output of one function becomes the input of another. We denote this composition as \[ (f \circ g)(x) = f(g(x)) \] where \(g(x)\) is the inner function and \(f(u)\) is the outer function.
In the given problem, the term \((x+2)^3\) exemplifies a composite function. Here, \(g(x) = x + 2 \) and \(f(u) = u^3 \). To differentiate this composite function, the chain rule was applied.
Steps involved:
In the given problem, the term \((x+2)^3\) exemplifies a composite function. Here, \(g(x) = x + 2 \) and \(f(u) = u^3 \). To differentiate this composite function, the chain rule was applied.
Steps involved:
- Differentiating the outer function \(u^3\) to get \(3u^2\).
- Substituting back the inner function \(u = x+2\), resulting in \(3(x+2)^2\).
- Finally, multiplying by the derivative of the inner function (which is \(1\)).
