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Last updated on July 21st, 2025

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Derivative of x/3

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We use the derivative of x/3, which is 1/3, 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/3 in detail.

Derivative of x/3 for Indian Students
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What is the Derivative of x/3?

We now understand the derivative of x/3. It is commonly represented as d/dx (x/3) or (x/3)', and its value is 1/3.

 

The function x/3 has a clearly defined derivative, indicating it is differentiable within its domain.

 

The key concepts are mentioned below: Linear Function: (f(x) = x/3).

 

Constant Rule: Rule for differentiating constant multiples of x.

 

Constant Function: The derivative of a constant is 0.

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Derivative of x/3 Formula

The derivative of x/3 can be denoted as d/dx (x/3) or (x/3)'.

 

The formula we use to differentiate x/3 is: d/dx (x/3) = 1/3 The formula applies to all x.

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Proofs of the Derivative of x/3

We can derive the derivative of x/3 using proofs. To show this, we will use the rule of differentiation for constant multiples.

 

There are several methods we use to prove this, such as: By First Principle Using Constant Rule Using Power Rule

 

We will now demonstrate that the differentiation of x/3 results in 1/3 using the above-mentioned methods: By First Principle The derivative of x/3 can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

 

To find the derivative of x/3 using the first principle, we will consider f(x) = x/3. 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/3, we write f(x + h) = (x + h)/3.

 

Substituting these into equation (1), f'(x) = limₕ→₀ [(x + h)/3 - x/3] / h = limₕ→₀ [h/3] / h = limₕ→₀ 1/3 Thus, f'(x) = 1/3. Hence, proved.

 

Using Constant Rule To prove the differentiation of x/3 using the constant rule, We use the formula: d/dx (c·x) = c Here, c = 1/3 Therefore, d/dx (x/3) = 1/3.

 

Using Power Rule We will now prove the derivative of x/3 using the power rule.

 

The step-by-step process is demonstrated below: Consider f(x) = x/3 = (1/3)x.

 

Using the power rule formula: d/dx [x^n] = n·x^(n-1) Here, n = 1. d/dx (x/3) = d/dx [(1/3)x] = (1/3)·d/dx [x^1] = (1/3)·1·x^(1-1) = 1/3

 

Thus, the derivative of x/3 is 1/3.

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Higher-Order Derivatives of x/3

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/3. 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/3, we generally use f^(n)(x) for the nth derivative of a function f(x) which tells us the change in the rate of change.

 

For linear functions like x/3, higher-order derivatives will be zero beyond the first derivative.

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Special Cases:

For any value of x, the derivative of x/3 is always 1/3.

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Common Mistakes and How to Avoid Them in Derivatives of x/3

Students frequently make mistakes when differentiating x/3.

 

These mistakes can be resolved by understanding the proper solutions.

 

Here are a few common mistakes and ways to solve them:

Mistake 1

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Not recognizing constant multiples

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Students may forget that when differentiating a constant multiple of x, like x/3, the constant remains unchanged as a factor in the derivative. Ensure to apply the constant rule correctly: d/dx (c·x) = c.

Mistake 2

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Incorrectly applying the power rule

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While differentiating linear functions, students might misapply the power rule.

 

Remember, for x^n, d/dx (x^n) = n·x^(n-1), but for x/3, it is (1/3)x^1, and the derivative is simply 1/3.

Mistake 3

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Neglecting to simplify

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Students often neglect to simplify the expression during differentiation, especially using the first principle. Ensure each step is simplified to avoid errors.

Mistake 4

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Overcomplicating simple derivatives

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There is a tendency to overcomplicate the process when differentiating simple linear functions.

 

Remember that linear functions have straightforward derivatives; for example, d/dx (x/3) is simply 1/3, without needing complex rules.

Mistake 5

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Misunderstanding higher-order derivatives

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Students might not realize that for linear functions, higher-order derivatives are zero beyond the first derivative. Understand that the first derivative of x/3 is 1/3, and further derivatives are 0.

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Examples Using the Derivative of x/3

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Problem 1

Calculate the derivative of (x/3)·(2x).

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Here, we have f(x) = (x/3)·(2x). Using the product rule, f'(x) = u′v + uv′ In the given equation, u = x/3 and v = 2x.

 

Let’s differentiate each term, u′ = d/dx (x/3) = 1/3 v′ = d/dx (2x) = 2 substituting into the given equation, f'(x) = (1/3)·(2x) + (x/3)·2 = (2/3)x + (2/3)x = 4x/3

 

Thus, the derivative of the specified function is 4x/3.

Explanation

We find the derivative of the given function by dividing the function into two parts.

 

The first step is finding its derivative and then combining them using the product rule to get the final result.

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Problem 2

A company tracks its revenue growth over time with the function y = x/3, where y represents the revenue in thousands and x is the time in months. If x = 12 months, calculate the rate of revenue growth.

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We have y = x/3 (revenue growth function)...(1) Now, we will differentiate the equation (1)

 

Take the derivative of x/3: dy/dx = 1/3

 

The rate of revenue growth is constant at 1/3 thousand dollars per month.

 

Given x = 12 months, the rate remains 1/3.

Explanation

We find that the revenue grows at a constant rate of 1/3 thousand dollars per month, which remains the same regardless of the time since the function is linear.

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Problem 3

Derive the second derivative of the function y = x/3.

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The first step is to find the first derivative, dy/dx = 1/3...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx (1/3) = 0 Therefore, the second derivative of the function y = x/3 is 0.

Explanation

We use the step-by-step process, where we start with the first derivative. The second derivative of a constant is always 0.

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Problem 4

Prove: d/dx ((x/3)²) = (2/3)x.

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Let’s start using the chain rule: Consider y = (x/3)² = (1/3)²x²

 

To differentiate, we use the chain rule: dy/dx = 2·(1/3)²·x = (2/9)x Thus, d/dx ((x/3)²) = (2/9)x.

Explanation

In this step-by-step process, we used the chain rule to differentiate the equation.

 

Then, we simplify the terms to find the final answer.

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Problem 5

Solve: d/dx (x/3x).

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To differentiate the function, we simplify first: d/dx (x/3x) = d/dx (1/3) Since the derivative of a constant is 0, = 0 Therefore, d/dx (x/3x) = 0.

Explanation

In this process, we simplify the given function to a constant and then differentiate, resulting in a zero derivative.

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FAQs on the Derivative of x/3

1.Find the derivative of x/3.

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2.Can we use the derivative of x/3 in real life?

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3.Is the derivative of x/3 the same at all points?

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4.What rule is used to differentiate x/3?

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5.Is the second derivative of x/3 always zero?

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Important Glossaries for the Derivative of x/3

  • 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 multiple of x is the constant itself.

 

  • Linear Function: A function of the form f(x) = mx + b, here m = 1/3, b = 0.

 

  • First Derivative: It is the initial result of a function, which gives us the rate of change of a specific function.

 

  • Second Derivative: The derivative of the first derivative, which indicates the rate of change of the rate of change. For linear functions, this is zero.
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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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Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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