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Magazine Chapter 2: Problem 15
Differentiate the function. \(R(a)=(3 a+1)^{2}\)
Short Answer
Expert verified
The derivative is \(18a + 6\).
Step by step solution
01
Recognize the Type of Function
The given function is \( R(a) = (3a+1)^2 \). This is a composite function where \( u = 3a + 1 \) and we have \( R(a) = u^2 \). To differentiate this, we will use the chain rule.
02
Differentiate the Outer Function
Differentiate the outer function \( u^2 \) with respect to \( u \). The derivative of \( u^2 \) with respect to \( u \) is \( 2u \).
03
Differentiate the Inner Function
Now find the derivative of the inner function \( u = 3a + 1 \) with respect to \( a \). The derivative of \( 3a + 1 \) is \( 3 \).
04
Apply the Chain Rule
Using the chain rule, the derivative of \( R(a) = (3a+1)^2 \) is the derivative of the outer function times the derivative of the inner function. This results in:\[\frac{dR}{da} = 2(3a + 1) \cdot 3 = 6(3a + 1)\]
05
Simplify the Result
Simplify the expression obtained:\[\frac{dR}{da} = 6(3a + 1) = 18a + 6\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Differentiation
Differentiation is a fundamental concept in calculus. It refers to the process of finding the derivative of a function, which represents the rate of change of the function with respect to a variable. In simpler terms, a derivative helps us understand how the output of a function changes as the input changes.
For instance, if we think of a function as describing the motion of a car on a road, its derivative tells us how fast the car is moving at each point in time.
Differentiating a function may seem daunting at first, but it gets easier with practice and understanding. The function in our exercise is a perfect example of using differentiation through the chain rule, as it involves a composite function.
For instance, if we think of a function as describing the motion of a car on a road, its derivative tells us how fast the car is moving at each point in time.
Differentiating a function may seem daunting at first, but it gets easier with practice and understanding. The function in our exercise is a perfect example of using differentiation through the chain rule, as it involves a composite function.
Composite Function
A composite function is essentially a function created by combining two or more functions. You can think of it as plugging one function into another. For example, if we have functions \(f(x)\) and \(g(x)\), a composite function can be represented as \(f(g(x))\).
In the exercise above, the function \(R(a) = (3a + 1)^2\) is a composite function. Here, the function \(3a + 1\) is nested inside the squaring function.
This composition is critical because it allows us to use the chain rule to differentiate. Understanding how to identify and treat composite functions helps simplify the process of taking derivatives significantly.
In the exercise above, the function \(R(a) = (3a + 1)^2\) is a composite function. Here, the function \(3a + 1\) is nested inside the squaring function.
This composition is critical because it allows us to use the chain rule to differentiate. Understanding how to identify and treat composite functions helps simplify the process of taking derivatives significantly.
Outer Function
In a composite function, the outer function is the operation applied to the inner function. It can be thought of as the 'outside' layer of a composition.
For our given function, \(R(a) = (3a + 1)^2\), the expression \(u^2\) is the outer function. Here, \(u\) is equivalent to \(3a + 1\).
The role of the outer function is significant in differentiation, especially when using the chain rule. We first differentiate the outer function with respect to its input, which provides a part of the formula for finding the derivative of the composite function.
For our given function, \(R(a) = (3a + 1)^2\), the expression \(u^2\) is the outer function. Here, \(u\) is equivalent to \(3a + 1\).
The role of the outer function is significant in differentiation, especially when using the chain rule. We first differentiate the outer function with respect to its input, which provides a part of the formula for finding the derivative of the composite function.
Inner Function
The inner function in a composite setup is the 'core' function that is nested inside the outer function.
For the function in our example, \(3a + 1\) represents the inner function. It's crucial to differentiate this component as part of the chain rule application.
When differentiating composite functions, we first recognize and differentiate the inner function. In our case, this differentiation yields a constant, \(3\), because the derivative of \(3a + 1\) with respect to \(a\) is \(3\). This step ensures that all parts of the composite function are accounted for in the differentiation process.
For the function in our example, \(3a + 1\) represents the inner function. It's crucial to differentiate this component as part of the chain rule application.
When differentiating composite functions, we first recognize and differentiate the inner function. In our case, this differentiation yields a constant, \(3\), because the derivative of \(3a + 1\) with respect to \(a\) is \(3\). This step ensures that all parts of the composite function are accounted for in the differentiation process.
