105 LearnersLast updated on July 22nd, 2025
Derivative of Negative x
We use the derivative of -x, which is -1, as a measuring tool for how the negative linear 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 in detail.
What is the Derivative of Negative x?
We now understand the derivative of -x. It is commonly represented as d/dx (-x) or (-x)', and its value is -1. The function -x 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 y = mx + b, where m and b are constants.
Constant Rule: The derivative of a constant is zero.
Negative Slope: A line with a negative slope descends from left to right.
Derivative of Negative x Formula
The derivative of -x can be denoted as d/dx (-x) or (-x)'. The formula we use to differentiate -x is: d/dx (-x) = -1 (or) (-x)' = -1 The formula applies to all x, as the derivative of a linear function is constant.
Proofs of the Derivative of Negative x
We can derive the derivative of -x using proofs. To show this, we will use basic differentiation rules. There are a few straightforward methods we use to prove this, such as:
By First Principle
The derivative of -x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.
To find the derivative of -x using the first principle, we will consider f(x) = -x. 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, we write f(x + h) = -(x + h).
Substituting these into equation (1), f'(x) = limₕ→₀ [-(x + h) + x] / h = limₕ→₀ [-h] / h = limₕ→₀ -1 f'(x) = -1. Hence, proved.
Using Basic Differentiation Rules
To prove the differentiation of -x using basic rules, consider the linear function y = -x.
The derivative of a constant times a function is the constant times the derivative of the function.
Thus, d/dx (-1 * x) = -1 * d/dx (x) The derivative of x with respect to x is 1. Therefore, d/dx (-x) = -1 * 1 = -1.
Higher-Order Derivatives of Negative x
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.
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).
For the nth Derivative of -x, we generally use fⁿ(x) for the nth derivative of a function f(x), but for a linear function like -x, all higher-order derivatives beyond the first are zero.
Special Cases:
For any real number x, the derivative of -x is always -1. The derivative remains constant at -1 regardless of the value of x.
Common Mistakes and How to Avoid Them in Derivatives of Negative x
Students frequently make mistakes when differentiating -x. These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
Confusing the Sign
Students may forget the negative sign and write the derivative as +1 instead of -1. Always remember that the derivative of a negative linear function is the negative of the slope.
Mistake 2
Misunderstanding Higher Derivatives
Some students may incorrectly assume that higher-order derivatives of linear functions are non-zero. Remember, for a linear function like -x, all higher-order derivatives beyond the first are zero.
Mistake 3
Applying Complex Rules Unnecessarily
While differentiating -x, students might unnecessarily apply complex differentiation rules like the product or quotient rule. For a simple linear function, using basic principles is sufficient. Stick to the basics to avoid unnecessary complications.
Mistake 4
Ignoring Constants and Coefficients
Students sometimes forget to multiply the constants placed before x. For example, they may incorrectly write d/dx (-5x) as -1. Ensure the derivative respects coefficients; the correct derivative is -5.
Mistake 5
Misapplying the Chain Rule
Students often incorrectly apply the chain rule to simple linear functions. For example: Incorrect: d/dx(-2x) = -2. Correct usage does not require the chain rule for linear derivatives: d/dx(-2x) = -2.
Examples Using the Derivative of Negative x
Problem 1
Calculate the derivative of (-x·3x).
Here, we have f(x) = -x·3x.
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = -x and v = 3x.
Let’s differentiate each term, u′= d/dx (-x) = -1 v′= d/dx (3x) = 3
Substituting into the given equation, f'(x) = (-1)(3x) + (-x)(3)
Let's simplify terms to get the final answer, f'(x) = -3x - 3x
Thus, the derivative of the specified function is -6x.
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 adjusts its pricing linearly, represented by y = -x, where y is the price adjustment and x is the time in months. If x = 5 months, find the rate of price change.
We have y = -x (price adjustment)...(1)
Now, we will differentiate equation (1)
Take the derivative of -x: dy/dx = -1 At x = 5 months, the rate of price change remains -1.
Explanation
We find that the rate of price change at any given month is constant at -1, indicating a consistent decrease in price over time.
Problem 3
Derive the second derivative of the function y = -x.
The first step is to find the first derivative, dy/dx = -1...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx (-1)
Since the derivative of a constant is zero, d²y/dx² = 0
Therefore, the second derivative of the function y = -x is 0.
Explanation
We use the step-by-step process, where we start with the first derivative and recognize that since the first derivative is a constant, the second derivative is zero.
Problem 4
Prove: d/dx (-x²) = -2x.
Let’s start by applying the power rule: Consider y = -x² The derivative of x² using the power rule is 2x.
Thus, d/dx (-x²) = -1 * d/dx (x²) = -1 * 2x = -2x
Hence proved.
Explanation
In this step-by-step process, we used the power rule to differentiate the equation. Then, we applied the negative sign to the result of the derivative.
Problem 5
Solve: d/dx (-x/x)
To differentiate the function, we simplify first: d/dx (-x/x) = d/dx (-1)
Since the derivative of a constant is zero, d/dx (-x/x) = 0
Therefore, the derivative of the simplified function is 0.
Explanation
In this process, we simplify the given function to recognize it as a constant and then differentiate, knowing the derivative of a constant is zero.
FAQs on the Derivative of Negative x
1.Find the derivative of -x.
2.Can we use the derivative of -x in real life?
3.What is the second derivative of -x?
4.How do you differentiate -x²?
5.Are the derivatives of -x and -1/x the same?
Important Glossaries for the Derivative of Negative x
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Linear Function: A function that forms a straight line, typically represented as y = mx + b.
- Constant Rule: A rule stating that the derivative of a constant is zero.
- Product Rule: A differentiation rule used for functions that are products of two functions.
- Higher-Order Derivative: Derivatives beyond the first, indicating further rates of change of a function.
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.
