Last updated on July 21st, 2025
We use the derivative of x/7, which is 1/7, 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/7 in detail.
We now understand the derivative of x/7. It is commonly represented as d/dx (x/7) or (x/7)', and its value is 1/7. The function x/7 has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Linear Function: x/7 is a simple linear function.
Constant Rule: The derivative of a constant times a function.
The derivative of x/7 can be denoted as d/dx (x/7) or (x/7)'. The formula we use to differentiate x/7 is: d/dx (x/7) = 1/7
We can derive the derivative of x/7 using proofs. To show this, we will use basic rules of differentiation. There are several methods we use to prove this, such as:
The derivative of x/7 can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.
To find the derivative of x/7 using the first principle, we will consider f(x) = x/7. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h
Given that f(x) = x/7,
we write f(x + h) = (x + h)/7.
Substituting these into the equation, f'(x) = limₕ→₀ [(x + h)/7 - x/7] / h = limₕ→₀ [h/7] / h = limₕ→₀ 1/7 f'(x) = 1/7
Hence, proved.
To prove the differentiation of x/7 using the constant multiplication rule: Consider f(x) = x/7 We use the formula d/dx (c * f(x)) = c * d/dx (f(x)), where c is a constant.
Here, c = 1/7 and f(x) = x.
Thus, d/dx (x/7) = 1/7 * d/dx (x) = 1/7 * 1 = 1/7
Hence, the derivative is 1/7.
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 for more complex functions, but for x/7, it is straightforward.
The first derivative is a constant, so all higher-order derivatives will be zero. 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/7, we generally use fⁿ(x). Since the first derivative is constant, all higher-order derivatives (second, third, etc.) are zero.
Since x/7 is a linear function with a constant slope, there are no points of discontinuity or undefined behavior. Therefore, there are no special cases where the derivative changes behavior within its domain.
Students frequently make mistakes when differentiating x/7. 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 (x/7·3).
Here, we have f(x) = x/7·3. Using the constant multiplication rule, f'(x) = 3·(1/7) f'(x) = 3/7 Thus, the derivative of the specified function is 3/7.
We find the derivative of the given function by recognizing it as a constant multiplication. The first step is finding its derivative by applying the constant multiplication rule to get the final result.
A company measures its profit as a function of production x, represented by y = x/7. If production is increased to 21 units, what is the rate of change of profit?
We have y = x/7 (profit function)...(1) Now, we will differentiate the equation (1). Take the derivative: dy/dx = 1/7
Given x = 21, the rate of change of profit is constant and equal to 1/7.
We find the rate of change of profit using the derivative, which is constant at 1/7, meaning that each additional unit produced increases the profit by 1/7 units.
Derive the second derivative of the function y = x/7.
The first step is to find the first derivative, dy/dx = 1/7... (1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [1/7] d²y/dx² = 0
Therefore, the second derivative of the function y = x/7 is 0.
We use the step-by-step process, where we start with the first derivative, which is a constant. Differentiating a constant gives zero; hence, the second derivative is 0.
Prove: d/dx (3x/7) = 3/7.
Let’s start using the constant multiplication rule: Consider y = 3x/7
To differentiate, dy/dx = 3 * d/dx (x/7)
Since the derivative of x/7 is 1/7, dy/dx = 3 * 1/7 dy/dx = 3/7
Hence proved.
In this step-by-step process, we used the constant multiplication rule to differentiate the equation. The constant factor is multiplied by the derivative of x/7 to derive the equation.
Solve: d/dx (x/7 + 2).
To differentiate the function, we separate the terms: d/dx (x/7 + 2) = d/dx (x/7) + d/dx (2)
We know d/dx (x/7) = 1/7 and d/dx (2) = 0 = 1/7 + 0
Therefore, d/dx (x/7 + 2) = 1/7
In this process, we differentiate each term separately using basic rules. The derivative of a constant is zero, and the derivative of x/7 is 1/7, leading to 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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