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101 LearnersLast updated on October 17, 2025
Derivative of 1/3
The derivative of a constant function like 1/3 is 0, which indicates that the function does not change as x changes. Derivatives are useful in determining rates of change in various contexts. We will now discuss the derivative of 1/3 in detail.
What is the Derivative of 1/3?
The derivative of a constant function, such as 1/3, is always 0. This is because a constant does not change regardless of the value of x, hence it has no rate of change.
The key concepts are as follows:
Constant Function: A function that always returns the same value, such as f(x) = 1/3.
Derivative of a Constant: The derivative of any constant is 0.
Derivative of 1/3 Formula
The derivative of a constant, including 1/3, can be denoted as d/dx (1/3) or (1/3)'.
The formula for differentiating any constant c is: d/dx (c) = 0
Therefore, d/dx (1/3) = 0.
Proofs of the Derivative of 1/3
We can prove that the derivative of 1/3 is 0 using basic calculus principles.
There are a few methods to demonstrate this:
By First Principle
The derivative of a constant can be shown using the First Principle, representing the derivative as the limit of the difference quotient. For a constant function f(x) = 1/3, the derivative is expressed as: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h = limₕ→₀ [1/3 - 1/3] / h = limₕ→₀ 0 / h = 0 Thus, the derivative of 1/3 is 0.
Using Constant Rule
The constant rule in differentiation states that the derivative of any constant is 0. Therefore, applying this rule: d/dx (1/3) = 0
Higher-Order Derivatives of 1/3
Higher-order derivatives of a constant, like 1/3, are also 0. When differentiating constants multiple times, you consistently get 0 at each step.
For example, the first derivative of 1/3 is 0, and the second derivative is also 0, and so on. This reflects the fact that constants do not change, hence their rates of change at any order are zero.
Special Cases
Since 1/3 is a constant, there are no special cases in differentiation because a constant does not depend on x.
The derivative remains 0 regardless of the value of x.
Common Mistakes and How to Avoid Them in Derivatives of 1/3
Students often make mistakes when dealing with derivatives of constants. These errors can be minimized by understanding the fundamental rules of differentiation. Here are some common mistakes and solutions:
Mistake 1
Mistaking Constants for Variables
Students sometimes confuse constants with variables, attempting to apply rules that involve variables.
Remember, the derivative of a constant like 1/3 is simply 0, without any additional calculations needed.
Mistake 2
Incorrect Application of Rules
Using differentiation rules meant for functions of x on constants is a common error.
Always apply the constant rule for differentiation, which states that the derivative of any constant is 0.
Mistake 3
Misunderstanding Higher-Order Derivatives
Some students incorrectly believe that differentiating a constant multiple times might yield non-zero values.
Remember, differentiating a constant any number of times always results in 0.
Mistake 4
Complicating Simple Problems
Overthinking the problem by attempting complex differentiation techniques on a constant can lead to confusion. Keep it simple: the derivative of a constant is always 0.
Mistake 5
Ignoring Simple Rules
Neglecting the basic rule that the derivative of a constant is 0 can lead to incorrect conclusions.
Always rely on basic differentiation rules for constants.
Examples Using the Derivative of 1/3
Problem 1
Calculate the derivative of the function f(x) = 1/3.
Since f(x) = 1/3 is a constant function, using the constant rule of differentiation: f'(x) = 0
Explanation
The derivative of a constant function, like 1/3, is always 0 because constants do not change with x.
Problem 2
A company produces widgets at a constant rate represented by f(x) = 1/3 widgets per hour. What is the rate of change of production?
The rate of change of production, represented by the derivative of f(x) = 1/3, is 0 widgets per hour.
Explanation
The production rate is constant, so it does not change with time; hence the derivative is 0.
Problem 3
Derive the second derivative of the function f(x) = 1/3.
The first derivative of f(x) = 1/3 is 0. Differentiating 0 gives us the second derivative: f''(x) = 0
Explanation
The higher-order derivatives of a constant function are 0 because constants do not vary with x.
Problem 4
Prove: d/dx (1/3 + x) = 1.
Let's differentiate the function f(x) = 1/3 + x: d/dx (1/3 + x) = d/dx (1/3) + d/dx (x) = 0 + 1 Thus, d/dx (1/3 + x) = 1.
Explanation
The derivative of 1/3 is 0 (since it's a constant), and the derivative of x is 1.
Adding these gives the result.
Problem 5
Solve: d/dx (1/3 - x).
To differentiate this function, use the basic rules: d/dx (1/3 - x) = d/dx (1/3) - d/dx (x) = 0 - 1 = -1
Explanation
The derivative of a constant (1/3) is 0, and the derivative of -x is -1, so the overall result is -1.
FAQs on the Derivative of 1/3
1.What is the derivative of 1/3?
2.Can the derivative of 1/3 be applied practically?
3.Is the derivative of 1/3 always zero?
4.What happens if you differentiate 1/3 multiple times?
5.Can I differentiate a function like 1/3 + x?
Important Glossaries for the Derivative of 1/3
- Derivative: The derivative of a function indicates how the function changes in response to a slight change in x.
- Constant Function: A function that returns the same value, such as f(x) = 1/3.
- First Derivative: The initial derivative of a function, representing its rate of change.
- Zero Derivative: The derivative of a constant function, indicating no change.
- Rate of Change: The change in a function's output in response to a change in input, often determined via derivatives.
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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.
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