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104 LearnersLast updated on October 3, 2025
Derivative of 1/3x
We use the derivative of 1/(3x), which is -1/(3x²), as a tool for understanding 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 discuss the derivative of 1/(3x) in detail.
What is the Derivative of 1/3x?
The derivative of 1/(3x) is commonly represented as d/dx (1/(3x)) or (1/(3x))', and its value is -1/(3x²). The function 1/(3x) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: -
Reciprocal Function: A function in the form of 1/f(x).
Power Rule: A basic rule of differentiation applicable to functions of the form x^n.
Constant Multiple Rule: Allows us to multiply the derivative of a function by a constant.
Derivative of 1/3x Formula
The derivative of 1/(3x) can be denoted as d/dx (1/(3x)) or (1/(3x))'.
The formula we use to differentiate 1/(3x) is: d/dx (1/(3x)) = -1/(3x²)
The formula applies to all x where x ≠ 0.
Proofs of the Derivative of 1/3x
We can derive the derivative of 1/(3x) using proofs. To show this, we will use algebraic manipulation along with the rules of differentiation. There are several methods we use to prove this, such as: -
- By First Principles
- Using Power Rule
- Using Constant Multiple Rule
We will now demonstrate that the differentiation of 1/(3x) results in -1/(3x²) using the above-mentioned methods:
By First Principles
The derivative of 1/(3x) can be proved using the First Principles, which express the derivative as the limit of the difference quotient. To find the derivative of 1/(3x) using the first principle, we will consider f(x) = 1/(3x).
Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)
Given that f(x) = 1/(3x), we write f(x + h) = 1/(3(x + h)).
Substituting these into equation (1), f'(x) = limₕ→₀ [1/(3(x + h)) - 1/(3x)] / h = limₕ→₀ [(3x - 3(x + h)) / (3x(3(x + h)))] / h = limₕ→₀ [-3h / (9x(x + h))]/ h = limₕ→₀ -3 / (9x(x + h))
As h approaches 0, f'(x) = -3 / (9x²) = -1 / (3x²)
Hence, proved.
Using Power Rule
The function 1/(3x) can be rewritten as\( (1/3)x^-1\).
Differentiating using the power rule, \(d/dx [(1/3)x^-1] = (1/3) * (-1)x^(-1-1) = -1/(3x²)\)
Using Constant Multiple Rule
We use the constant multiple rule to differentiate 1/(3x).
Consider g(x) = 1/x, which has a derivative of -1/x².
Then, f(x) = 1/(3x) = (1/3)g(x).
Using the constant multiple rule, f'(x) = (1/3) * g'(x) = (1/3) * (-1/x²) = -1/(3x²).
Higher-Order Derivatives of 1/3x
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 help us understand functions like 1/(3x).
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 1/(3x), we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change.
Special Cases:
1. When x is 0, the derivative is undefined because 1/(3x) is undefined at x = 0. 2.
When x is 1, the derivative of 1/(3x) = -1/(3 * 1²), which is -1/3.
Common Mistakes and How to Avoid Them in Derivatives of 1/3x
Students frequently make mistakes when differentiating 1/(3x). These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
Not simplifying the equation
Students may forget to simplify the equation, which can lead to incomplete or incorrect results. They often skip steps and directly arrive at the result, especially when using power or product rules. Ensure that each step is written in order. Students might think it is tedious, but it is important to avoid errors in the process.
Mistake 2
Forgetting the Undefined Points
They might not remember that 1/(3x) is undefined at the point x = 0. Keep in mind that you should consider the domain of the function that you differentiate. It will help you understand that the function is not continuous at such certain points.
Mistake 3
Incorrect use of Power Rule
While differentiating functions such as 1/(3x), students may misapply the power rule. For example: Incorrect differentiation: d/dx (1/(3x)) = -1/x. Correct application: d/dx (x^-1) = -1x^-2. To avoid this mistake, ensure you apply the power rule correctly and simplify appropriately.
Mistake 4
Not writing Constants and Coefficients
There is a common mistake that students at times forget to consider the constants placed before 1/x. For example, they incorrectly write d/dx (1/(3x)) = -1/x².
Students should check the constants in the terms and ensure they are multiplied properly. For example, the correct equation is d/dx (1/(3x)) = -1/(3x²).
Mistake 5
Not Applying the Constant Rule
Students often forget to use the constant rule. This happens when the constant factor in the function is ignored. For example: Incorrect: d/dx (1/(3x)) = -1/x².
To fix this error, students should include the constant factor when differentiating, using the rule that multiplying by a constant factor affects the derivative.
Examples Using the Derivative of 1/3x
Problem 1
Calculate the derivative of (1/(3x) * ln(x))
Here, we have f(x) = 1/(3x) * ln(x). Using the product rule, f'(x) = u′v + uv′ In the given equation, u = 1/(3x) and v = ln(x).
Let’s differentiate each term, u′ = d/dx (1/(3x)) = -1/(3x²) v′ = d/dx (ln(x)) = 1/x
Substituting into the given equation, f'(x) = (-1/(3x²)) * ln(x) + (1/(3x)) * (1/x)
Let’s simplify terms to get the final answer, f'(x) = -ln(x)/(3x²) + 1/(3x²)
Thus, the derivative of the specified function is -ln(x)/(3x²) + 1/(3x²).
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.
Problem 2
A company uses the function y = 1/(3x) to model the cost per unit of production, where x is the number of units produced. If x = 2 units, calculate the rate of change of cost per unit.
We have y = 1/(3x) (cost per unit)...(1)
Now, we will differentiate the equation (1) Take the derivative of 1/(3x): dy/dx = -1/(3x²) Given x = 2 (substitute this into the derivative) dy/dx = -1/(3 * 2²) = -1/12
Hence, the rate of change of cost per unit at x = 2 is -1/12.
Explanation
We find the rate of change of cost per unit at x = 2 as -1/12, which means that the cost per unit will decrease at this rate as production increases.
Problem 3
Derive the second derivative of the function y = 1/(3x).
The first step is to find the first derivative, dy/dx = -1/(3x²)...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-1/(3x²)] = -1 * [d/dx (x-2)] * (1/3) = -1 * (-2x-3) * (1/3) = 2/(3x³)
Therefore, the second derivative of the function y = 1/(3x) is 2/(3x³).
Explanation
We use the step-by-step process, where we start with the first derivative. Using the power rule, we differentiate
x-2. We then simplify the terms to find the final answer.
Problem 4
Prove: d/dx (1/(3x²)) = -2/(3x³).
Let’s start using the power rule: Consider y = 1/(3x²) = (1/3)x^-2 To differentiate, we use the power rule:
dy/dx = (1/3) * (-2)x(-2-1) = -2/(3x³)
Hence proved.
Explanation
In this step-by-step process, we used the power rule to differentiate the equation. Then, we replaced the expression with its derivative. As a final step, we simplified the terms to derive the equation.
Problem 5
Solve: d/dx ((1/(3x)) * e^x)
To differentiate the function, we use the product rule:\( d/dx ((1/(3x)) * e^x) = (d/dx (1/(3x)) * e^x + 1/(3x) * d/dx(e^x))\)
We will substitute \(d/dx (1/(3x)) = -1/(3x²) and d/dx (e^x) = e^x = (-1/(3x²)) * e^x + (1/(3x)) * e^x = e^x [-1/(3x²) + 1/(3x)] = e^x [(-1 + x)/(3x²)]\)
Therefore, d/dx \(((1/(3x)) * e^x) = e^x [(-1 + x)/(3x²)]\)
Explanation
In this process, we differentiate the given function using the product rule. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of 1/3x
1.Find the derivative of 1/3x.
2.Can we use the derivative of 1/3x in real life?
3.Is it possible to take the derivative of 1/3x at the point where x = 0?
4.What rule is used to differentiate (1/3x) * ln(x)?
5.Are the derivatives of 1/3x and 3/x the same?
Important Glossaries for the Derivative of 1/3x
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Reciprocal Function: A function in the form of 1/f(x), such as 1/(3x).
- Power Rule: A basic rule of differentiation applicable to functions of the form x^n.
- Constant Multiple Rule: A rule allowing the multiplication of the derivative of a function by a constant.
- Undefined: A term used to describe a function or expression that has no meaning at a certain point or value of x.
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.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
