Problem 29
Question
Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Sum. $$y=a x+b$$
Step-by-Step Solution
Verified Answer
The derivative of the function $$y = ax + b$$ is $$y' = a$$.
1Step 1: Identify the Function and Rules for Differentiation
Recognize that the function to be differentiated is a linear function of the form: $$y=ax+b$$ For this function, the derivative can be found by applying the power rule, which states that the derivative of a function in the form of $$x^n$$ is $$nx^{n-1}$$, and that the derivative of a constant is zero.
2Step 2: Apply the Power Rule
Differentiate the first term, $$ax$$, by applying the power rule. Since $$x$$ is actually $$x^1$$, its derivative will be $$a \times 1 \times x^{1-1} = a$$.The second term, $$b$$, is a constant and its derivative is zero.
3Step 3: Combine the Results
Combine the derivatives of individual terms to find the derivative of the whole function. The derivative of the constant term $$b$$ is zero, so it does not appear in the final derivative. Therefore, the derivative of the function $$y = ax + b$$ is just $$a$$.
Key Concepts
Power Rule for DifferentiationDeriving ConstantsLinear Function Differentiation
Power Rule for Differentiation
Understanding the power rule for differentiation is a crucial skill in calculus. It simplifies the process of finding the derivative of a function that can be expressed as a power of a variable. The power rule states that if you have a function of the form \( f(x) = x^n \), where \( n \) is any real number, then the derivative of that function with respect to \( x \) is \( f'(x) = nx^{n-1} \).
For example, if you have the function \( f(x) = x^3 \), applying the power rule would give you \( f'(x) = 3x^{3-1} = 3x^2 \). It's easy to see how this rule streamlines the differentiation process, especially when dealing with higher powers of \( x \). Moreover, when \( n=1 \), which is often the case in linear functions, we can deduce that the derivative simplifies to the coefficient in front of \( x \), since \( x^{1-1} = x^0 = 1 \), leaving us with just the coefficient as the derivative.
For example, if you have the function \( f(x) = x^3 \), applying the power rule would give you \( f'(x) = 3x^{3-1} = 3x^2 \). It's easy to see how this rule streamlines the differentiation process, especially when dealing with higher powers of \( x \). Moreover, when \( n=1 \), which is often the case in linear functions, we can deduce that the derivative simplifies to the coefficient in front of \( x \), since \( x^{1-1} = x^0 = 1 \), leaving us with just the coefficient as the derivative.
Deriving Constants
When working with functions in calculus, it's essential to remember that the derivative of a constant is zero. This principle stems from the concept that a constant does not change, hence it has no rate of change, which is precisely what a derivative measures. In mathematical terms, if \( c \) is a constant, then \( \frac{d}{dx}c = 0 \).
This concept is particularly relevant when differentiating polynomials or linear functions which include both variable terms and constant terms. For instance, given a linear function like \( y = 3x + 5 \), the derivative of the constant term \( 5 \) with respect to \( x \) is \( 0 \), and so it effectively 'disappears' in the derivative of the function. Understanding this aspect of differentiation eliminates confusion and simplifies computation by allowing us to focus only on the parts of a function that do change.
This concept is particularly relevant when differentiating polynomials or linear functions which include both variable terms and constant terms. For instance, given a linear function like \( y = 3x + 5 \), the derivative of the constant term \( 5 \) with respect to \( x \) is \( 0 \), and so it effectively 'disappears' in the derivative of the function. Understanding this aspect of differentiation eliminates confusion and simplifies computation by allowing us to focus only on the parts of a function that do change.
Linear Function Differentiation
Linear functions, which have the form \( y = ax + b \), where \( a \) and \( b \) are constants, are perhaps the simplest to differentiate. The differentiation of linear functions directly applies the two concepts we've discussed: the power rule and deriving constants.
To find the derivative of \( y = ax + b \), you apply the power rule to the term \( ax \) (since it is equivalent to \( ax^1 \)) giving us a derivative of \( a \times 1 \times x^{1-1} = a \times 1 \times 1 = a \). The term \( b \), being a constant, has a derivative of zero. Therefore, combining these two results, the derivative of the entire linear function simplifies to \( y' = a \). Remember, the derivative of a linear equation represents the slope, or the rate of change, which in a linear function, remains constant. This inherent property of linear functions underscores why their differentiation is straightforward and consistent.
To find the derivative of \( y = ax + b \), you apply the power rule to the term \( ax \) (since it is equivalent to \( ax^1 \)) giving us a derivative of \( a \times 1 \times x^{1-1} = a \times 1 \times 1 = a \). The term \( b \), being a constant, has a derivative of zero. Therefore, combining these two results, the derivative of the entire linear function simplifies to \( y' = a \). Remember, the derivative of a linear equation represents the slope, or the rate of change, which in a linear function, remains constant. This inherent property of linear functions underscores why their differentiation is straightforward and consistent.
Other exercises in this chapter
Problem 28
Find the derivative. $$\frac{d}{d x}(3 x+2)$$
View solution Problem 29
Write the differential \(d y\) for each function. $$y=x^{3}$$
View solution Problem 29
$$y=x(x+1)^{2}(x-2)^{3}$$
View solution Problem 29
Find the derivative of the function \(y=\left(4.82 x^{2}-8.25 x\right)^{3}\) when \(x=3.77\).
View solution