Last updated on July 22nd, 2025
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.
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.
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
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:
We will now demonstrate that the differentiation of x-4 results in 1 using the above-mentioned methods:
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.
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.
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.
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.
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:
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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