102 LearnersLast updated on August 5th, 2025
Derivative of x/5
We use the derivative of x/5, which is 1/5, to understand how the function changes with respect to a slight change in x. Derivatives can assist us in calculating profit or loss in real-life situations. We will now discuss the derivative of x/5 in detail.
What is the Derivative of x/5?
We now understand the derivative of x/5. It is commonly represented as d/dx (x/5) or (x/5)', and its value is 1/5.
The function x/5 has a constant derivative, indicating it is differentiable across its entire domain.
The key concepts include: Linear Function: A function of the form f(x) = mx + b, where m and b are constants.
Constant Rule: The rule for differentiating constant multiples of functions.
Slope: The measure of steepness of a line, which in this case is the constant 1/5.
Derivative of x/5 Formula
The derivative of x/5 can be denoted as d/dx (x/5) or (x/5)'.
The formula we use to differentiate x/5 is: d/dx (x/5) = 1/5 (or) (x/5)' = 1/5
The formula applies to all x in the real number set.
Proofs of the Derivative of x/5
We can derive the derivative of x/5 using proofs. To show this, we will use basic differentiation rules.
There are several methods we use to prove this, such as:
By First Principle Using the Constant Rule We will now demonstrate that the differentiation of x/5 results in 1/5 using these methods:
By First Principle The derivative of x/5 can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.
To find the derivative of x/5 using the first principle, we consider f(x) = x/5.
Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = x/5, we write f(x + h) = (x + h)/5.
Substituting these into equation (1), f'(x) = limₕ→₀ [(x + h)/5 - x/5] / h = limₕ→₀ [x/5 + h/5 - x/5] / h = limₕ→₀ [h/5] / h = limₕ→₀ 1/5 Therefore, f'(x) = 1/5.
Hence, proved.
Using the Constant Rule To prove the differentiation of x/5 using the constant rule, We recognize that x/5 is a linear function with a slope of 1/5. By the constant multiple rule: d/dx (c·f(x)) = c·d/dx (f(x)) Let f(x) = x and c = 1/5, d/dx (x/5) = 1/5·d/dx (x) Since d/dx (x) = 1, d/dx (x/5) = 1/5·1 = 1/5.
Higher-Order Derivatives of x/5
When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can provide additional insights into the behavior of functions.
For the function x/5, which is linear, the first derivative is 1/5, and all higher-order derivatives are 0.
This indicates that the function's rate of change is constant, and there are no changes in the rate of change.
Special Cases:
The derivative of x/5 is constant and defined everywhere on the real line. The derivative does not change with the value of x, making it straightforward to compute.
Common Mistakes and How to Avoid Them in Derivatives of x/5
Students frequently make mistakes when differentiating constant multiples of x. 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 slope
Students may forget that the derivative of x/5 is a constant 1/5.
Sometimes they overcomplicate the process by trying to apply rules meant for more complex functions.
Remember that the derivative of a linear function ax + b is simply a.
Mistake 2
Misapplying the Constant Rule
They might misapply the constant rule, thinking that constants affect the differentiation process more than they do.
Keep in mind that for a function like x/5, the constant only serves to multiply the derivative of x, which is 1.
Mistake 3
Forgetting the Domain of Linear Functions
While differentiating, students might forget that the domain of x/5 is all real numbers.
Remember that linear functions are defined everywhere, so their derivatives are as well.
Mistake 4
Incorrectly using the First Principle
Students may incorrectly set up the limit process in the first principle for finding derivatives.
Ensure that the limit is correctly established and that terms cancel properly to avoid errors.
Mistake 5
Not writing Constants and Coefficients
There is a common mistake where students forget to multiply by constants correctly.
For example, they might incorrectly write d/dx (3x/5) as 1/5. The correct derivative is 3/5.
Examples Using the Derivative of x/5
Problem 1
Calculate the derivative of (x/5 + 7).
Here, we have f(x) = x/5 + 7. The derivative of a constant is 0, so we only differentiate x/5.
Using basic rules of differentiation, f'(x) = d/dx (x/5) + d/dx (7) = 1/5 + 0
Therefore, the derivative of the specified function is 1/5.
Explanation
We find the derivative of the given function by differentiating each term separately.
The derivative of a constant is 0, and the derivative of x/5 is 1/5.
Problem 2
A company charges a rate represented by the function y = x/5 dollars per hour for a service. If x = 10 hours, calculate the rate of change of the charge with respect to time.
We have y = x/5 (rate of charge)...(1) Now, we will differentiate the equation (1)
Take the derivative of x/5: dy/dx = 1/5 This indicates the rate of change is constant at 1/5 dollars per hour.
Given x = 10, the rate of change remains 1/5 dollars per hour.
Explanation
We find the rate of change of charge with respect to time as constant, meaning the cost increases steadily at 1/5 dollars per hour regardless of the number of hours.
Problem 3
Derive the second derivative of the function y = x/5.
The first step is to find the first derivative, dy/dx = 1/5...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx (1/5) Since 1/5 is a constant, d²y/dx² = 0
Therefore, the second derivative of the function y = x/5 is 0.
Explanation
We use the step-by-step process, starting with the first derivative.
Since the first derivative is a constant, the second derivative is 0, indicating no change in the rate of change.
Problem 4
Prove: d/dx (3x/5) = 3/5.
Consider y = 3x/5 To differentiate, we use the constant multiple rule: dy/dx = 3·d/dx (x/5) Since d/dx (x/5) = 1/5, dy/dx = 3·1/5 dy/dx = 3/5 Hence proved.
Explanation
In this step-by-step process, we used the constant multiple rule to differentiate the equation. We replaced x/5 with its derivative and computed the result.
Problem 5
Solve: d/dx (x/5 + x²).
To differentiate the function, we use basic rules of differentiation: d/dx (x/5 + x²) = d/dx (x/5) + d/dx (x²)
The derivative of x/5 is 1/5, and the derivative of x² is 2x. = 1/5 + 2x Therefore, d/dx (x/5 + x²) = 1/5 + 2x.
Explanation
In this process, we differentiate each term of the given function separately and then combine them to obtain the final result.
FAQs on the Derivative of x/5
1.Find the derivative of x/5.
2.Can we use the derivative of x/5 in real life?
3.Is the derivative of x/5 affected by changes in x?
4.What rule is used to differentiate x/5?
5.Are the derivatives of x/5 and 5x the same?
Important Glossaries for the Derivative of x/5
- 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 m and b are constants.
- Constant Rule: A rule stating that the derivative of a constant multiplied by a function is the constant times the derivative of the function.
- First Derivative: The initial result of differentiating a function, which gives us the rate of change.
- Constant: A value that does not change, used in the context of the multiple and constant terms in differentiation.
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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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