104 LearnersLast updated on July 22nd, 2025
Derivative of x-4
We use the derivative of x-4, which is 1, as a measuring tool for how the function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now talk about the derivative of x-4 in detail.
What is the Derivative of x-4?
We now understand the derivative of x-4. It is commonly represented as d/dx (x-4) or (x-4)', and its value is 1. The function x-4 has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Linear Function: A function of the form f(x) = mx + b.
Constant Rule: The derivative of a constant is 0.
Power Rule: If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹.
Derivative of x-4 Formula
The derivative of x-4 can be denoted as d/dx (x-4) or (x-4)'. The formula we use to differentiate x-4 is: d/dx (x-4) = 1 (or) (x-4)' = 1
Proofs of the Derivative of x-4
We can derive the derivative of x-4 using proofs. To show this, we will use differentiation rules. There are several methods we use to prove this, such as:
- Using the Power Rule
- Using the Constant Rule
We will now demonstrate that the differentiation of x-4 results in 1 using the above-mentioned methods:
Using the Power Rule
The derivative of x-4 can be proved using the power rule, which expresses the derivative as the power of x reduced by one.
To find the derivative of x-4, we will consider f(x) = x-4. Its derivative can be expressed as: f'(x) = d/dx (x-4) = d/dx (x) - d/dx (4) = 1 - 0 Hence, f'(x) = 1. Hence, proved.
Using the Constant Rule
To prove the differentiation of x-4 using the constant rule, We use the formula: d/dx (c) = 0, where c is a constant.
Consider f(x) = x-4 f'(x) = d/dx (x-4) = d/dx (x) - d/dx (4) As per the constant rule, f'(x) = 1 - 0 = 1.
Thus, the derivative of x-4 is 1.
Higher-Order Derivatives of x-4
When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.
To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like x-4.
For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′ (x) Similarly, the third derivative, f′′′(x) is the result of the second derivative and this pattern continues.
For the nth Derivative of x-4, we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change. For x-4, all higher-order derivatives are 0.
Special Cases:
The derivative of x-4 is always 1 because it is a linear function. There are no points at which the derivative of x-4 is undefined.
Common Mistakes and How to Avoid Them in Derivatives of x-4
Students frequently make mistakes when differentiating x-4. These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
Not simplifying the equation
Students may forget to simplify the equation, which can lead to incomplete or incorrect results. They often skip steps and directly arrive at the result, especially when using rules like the constant rule. Ensure that each step is written in order. Students might think it is awkward, but it is important to avoid errors in the process.
Mistake 2
Confusing the Derivative with the Function
They might confuse the function x-4 with its derivative. Keep in mind that the derivative is the rate of change, not the function itself. This will help you understand that the derivative of x-4 is always 1.
Mistake 3
Incorrect application of rules
While differentiating functions like x-4, students misapply rules such as the power rule. For example: Incorrect differentiation: d/dx (x-4) = 0. The correct application is: d/dx (x) = 1 and d/dx (-4) = 0. To avoid this mistake, write the rules without errors. Always check for errors in the calculation and ensure it is properly simplified.
Mistake 4
Not writing Constants and Coefficients
There is a common mistake that students at times forget to apply the coefficient rule properly. For example, they incorrectly write d/dx (5(x-4)) = 1. Students should check the constants in the terms and ensure they are multiplied properly. For example, the correct equation is d/dx (5(x-4)) = 5.
Mistake 5
Confusion with Chain Rule
Students often incorrectly think they need to apply the chain rule. This happens when the function x-4 is mistaken as a composition. To fix this error, students should recognize that x-4 is a simple linear function. The derivative is straightforward, d/dx (x-4) = 1.
Examples Using the Derivative of x-4
Problem 1
Calculate the derivative of 2(x-4).
Here, we have f(x) = 2(x-4). Using the constant multiple rule, f'(x) = 2 d/dx (x-4) In the given equation, the derivative of x-4 is 1. Thus, f'(x) = 2 * 1 f'(x) = 2 Thus, the derivative of the specified function is 2.
Explanation
We find the derivative of the given function by applying the constant multiple rule. The first step is finding the derivative of x-4, which is 1, and then multiplying by the constant to get the final result.
Problem 2
A car is moving along a straight road. The position of the car is given by the function s = x-4, where s represents the position and x is time in seconds. Find the velocity of the car.
We have s = x-4 (position of the car)...(1) Now, we will differentiate the equation (1) Take the derivative of x-4: ds/dx = 1 Hence, the velocity of the car is 1 meter per second.
Explanation
We find the velocity of the car by differentiating its position function. The derivative of x-4 is 1, which means the car moves at a constant velocity of 1 meter per second.
Problem 3
Derive the second derivative of the function y = x-4.
The first step is to find the first derivative, dy/dx = 1...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [1]
Since the derivative of a constant is 0, d²y/dx² = 0
Therefore, the second derivative of the function y = x-4 is 0.
Explanation
We use the step-by-step process, where we start with the first derivative. Since the derivative of a constant is 0, the second derivative is simply 0.
Problem 4
Prove: d/dx (x²-4x) = 2x-4.
Let’s start using the power rule: Consider y = x²-4x
To differentiate, we use the power rule: dy/dx = d/dx (x²) - d/dx (4x) = 2x - 4
Hence, the derivative is 2x-4.
Explanation
In this step-by-step process, we used the power rule to differentiate each term in the equation. The derivative of x² is 2x, and the derivative of -4x is -4, giving us the final result.
Problem 5
Solve: d/dx ((x-4)/x).
To differentiate the function, we use the quotient rule: d/dx ((x-4)/x) = (d/dx (x-4).x - (x-4).d/dx(x))/x²
We will substitute d/dx (x-4) = 1 and d/dx (x) = 1 = (1.x - (x-4).1)/x² = (x - x + 4)/x² = 4/x²
Therefore, d/dx ((x-4)/x) = 4/x²
Explanation
In this process, we differentiate the given function using the quotient rule. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of x-4
1.Find the derivative of x-4.
2.Can we use the derivative of x-4 in real life?
3.Is it possible to take the derivative of x-4 at any point?
4.What rule is used to differentiate (x-4)/x?
5.Are the derivatives of x-4 and (x-4)² the same?
6.Can we find the derivative of the x-4 formula?
Important Glossaries for the Derivative of x-4
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Linear Function: A function of the form f(x) = mx + b, where the graph is a straight line.
- Constant Rule: The derivative of a constant is zero. Power Rule: If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹.
- Quotient Rule: A rule for differentiating functions that are divided by each other. ```
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
