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101 LearnersLast updated on October 17, 2025
Derivative of x/6
We use the derivative of x/6, which is 1/6, as a measuring tool for how the function x/6 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/6 in detail.
What is the Derivative of x/6?
We now understand the derivative of x/6. It is commonly represented as d/dx (x/6) or (x/6)', and its value is 1/6. The function x/6 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) = ax + b.
Constant Rule: The rule for differentiating a constant times a function.
Derivative of x: The derivative of x with respect to x is 1.
Derivative of x/6 Formula
The derivative of x/6 can be denoted as d/dx (x/6) or (x/6)'.
The formula we use to differentiate x/6 is: d/dx (x/6) = 1/6
The formula applies to all real numbers x.
Proofs of the Derivative of x/6
We can derive the derivative of x/6 using proofs. To show this, we will use basic rules of differentiation.
There are several methods we use to prove this, such as:
By Constant Rule
The derivative of x/6 can be proved using the Constant Rule, which states that the derivative of a constant times a function is the constant times the derivative of the function. To find the derivative of x/6, we consider f(x) = x/6. Its derivative can be expressed as: f'(x) = d/dx (1/6 * x) = 1/6 * d/dx (x) = 1/6 * 1 = 1/6 Hence, proved.
Using the Quotient Rule
To prove the differentiation of x/6 using the quotient rule, We represent x/6 as x divided by 6, which is a constant. Let u = x and v = 6. By quotient rule: d/dx [u/v] = [v * u' - u * v'] / v² Substituting u = x and v = 6, and knowing v' = 0 because v is constant, d/dx (x/6) = [6 * 1 - x * 0] / 6² = 6 / 36 = 1/6 Hence, proved.
Higher-Order Derivatives of x/6
When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives of linear functions like x/6 can be straightforward. 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 for linear functions are zero beyond the first derivative.
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 and is denoted using f′′(x).
For linear functions like x/6, the second derivative is 0. Similarly, the third derivative, f′′′(x), is also 0, and this pattern continues.
Special Cases:
The derivative of x/6 is always 1/6, regardless of the value of x.
Because it is a linear function, there are no points at which the derivative is undefined.
Common Mistakes and How to Avoid Them in Derivatives of x/6
Students frequently make mistakes when differentiating x/6. These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
Not recognizing the constant coefficient
Students may forget that 1/6 is a constant coefficient.
They might incorrectly try to differentiate it further. Remember, constants remain constants when differentiating linear functions.
Mistake 2
Misapplying rules of differentiation
Students might incorrectly use differentiation rules like the product or quotient rule when not necessary.
For x/6, the constant rule suffices since 1/6 is a constant.
Mistake 3
Incorrectly simplifying the derivative
While differentiating simple functions like x/6, students might overcomplicate the process by introducing unnecessary steps.
Remember, the derivative of x/6 is straightforward: 1/6.
Mistake 4
Forgetting the basic derivative of x
There is a common mistake that students sometimes forget the basic derivative of x, which is 1.
Always start with known basic derivatives when solving problems.
Mistake 5
Overcomplicating higher-order derivatives
Students often overcomplicate higher-order derivatives of linear functions.
For x/6, remember that all higher-order derivatives beyond the first are 0.
Examples Using the Derivative of x/6
Problem 1
Calculate the derivative of (x/6 + 5).
Here, we have f(x) = x/6 + 5. Using the sum rule of derivatives, f'(x) = d/dx (x/6) + d/dx (5) = 1/6 + 0 Thus, the derivative of the specified function is 1/6.
Explanation
We find the derivative of the given function by applying the sum rule and recognizing that the derivative of a constant is 0. This simplifies the derivative to 1/6.
Problem 2
The length of a line segment is represented by the function y = x/6, where x represents the change in a certain dimension. If x = 12 units, find the rate of change of the length.
We have y = x/6 (length of the line segment)...(1) Now, we will differentiate the equation (1) Take the derivative of x/6: dy/dx = 1/6 The rate of change is constant and equal to 1/6, regardless of the value of x.
Explanation
The rate of change of the length is constant because the derivative of x/6 is a constant, 1/6.
This means the length increases at a consistent rate as x changes.
Problem 3
Derive the second derivative of the function y = x/6.
The first step is to find the first derivative, dy/dx = 1/6...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx (1/6) Since 1/6 is a constant, its derivative is 0. Therefore, the second derivative of the function y = x/6 is 0.
Explanation
Since the first derivative of a linear function is constant, the second derivative of the function is 0, indicating no further rate of change.
Problem 4
Prove: d/dx (x²/6) = x/3.
Let’s use the power rule: Consider y = x²/6 To differentiate, apply the power rule: dy/dx = (2/6)x^(2-1) = (1/3)x Thus, d/dx (x²/6) = x/3.
Explanation
In this step-by-step process, we used the power rule to differentiate the quadratic term.
By simplifying, we found the derivative to be x/3.
Problem 5
Solve: d/dx ((x+1)/6)
To differentiate the function, we use the constant multiple rule: d/dx ((x+1)/6) = 1/6 * d/dx (x+1) = 1/6 * (1 + 0) = 1/6 Therefore, d/dx ((x+1)/6) = 1/6
Explanation
In this process, we differentiate the given function using the constant multiple rule.
The derivative of a constant remains unchanged.
FAQs on the Derivative of x/6
1.Find the derivative of x/6.
2.Can we use the derivative of x/6 in real life?
3.Is it possible to take the derivative of x/6 at any point?
4.What rule is used to differentiate x/6?
5.Are the derivatives of x/6 and (x/6)² the same?
6.Can we find the derivative of the x/6 formula?
Important Glossaries for the Derivative of x/6
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Constant Rule: A rule stating that the derivative of a constant times a function is the constant times the derivative of the function.
- Linear Function: A function of the form f(x) = ax + b, where a and b are constants.
- First Derivative: The initial result of differentiating a function, indicating the rate of change.
- Higher-Order Derivatives: Derivatives of a function beyond the first derivative, indicating further changes in rates. ```
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Jaskaran Singh Saluja
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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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: He loves to play the quiz with kids through algebra to make kids love it.
