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108 LearnersLast updated on September 9, 2025
Derivative of -x²
We use the derivative of -x², which is -2x, 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 talk about the derivative of -x² in detail.
What is the Derivative of -x²?
We now understand the derivative of -x².
It is commonly represented as d/dx (-x²) or (-x²)', and its value is -2x.
The function -x² has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Polynomial Function: A function like -x², which is a basic polynomial.
Power Rule: A rule used for differentiating polynomial functions.
Derivative of -x² Formula
The derivative of -x² can be denoted as d/dx (-x²) or (-x²)'. The formula we use to differentiate -x² is: d/dx (-x²) = -2x The formula applies to all x in the real number domain.
Proofs of the Derivative of -x²
We can derive the derivative of -x² using proofs.
To show this, we will use the rules of differentiation.
There are several methods we use to prove this, such as:
Using the Power Rule
Using the First Principle
Using the Power Rule
To prove the differentiation of -x² using the power rule,
We use the formula: d/dx (x^n) = n*x^(n-1)
For our function, n = 2 and the coefficient is -1: d/dx (-x²) = -1 * 2 * x^(2-1) = -2x
Hence, proved.
Using the First Principle
The derivative of -x² can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.
To find the derivative of -x² using the first principle, we will consider f(x) = -x².
Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)
Given that f(x) = -x², we write f(x + h) = -(x + h)².
Substituting these into equation (1), f'(x) = limₕ→₀ [-(x + h)² + x²] / h = limₕ→₀ [-(x² + 2xh + h²) + x²] / h = limₕ→₀ [-2xh - h²] / h = limₕ→₀ [-2x - h] As h approaches 0, the expression simplifies to: f'(x) = -2x
Hence, proved.
Higher-Order Derivatives of -x²
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 -x².
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 -x², we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change. (continuing for higher-order derivatives).
Special Cases:
When x is 0, the derivative of -x² is -2(0), which is 0.
Common Mistakes and How to Avoid Them in Derivatives of -x²
Students frequently make mistakes when differentiating -x². 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 solving using rules for differentiation. Ensure that each step is written in order. Students might think it is awkward, but it is important to avoid errors in the process.
Mistake 2
Forgetting Negative Signs
Students might not apply the negative sign correctly when differentiating. The derivative of -x² is -2x, not 2x. Make sure to maintain the negative sign throughout the differentiation process.
Mistake 3
Incorrect Use of the Power Rule
While differentiating functions such as -x², students sometimes misapply the power rule.
For example, they might incorrectly write the derivative as -x instead of -2x. To avoid this mistake, write the power rule without errors. Always check for errors in the calculation and ensure it is properly simplified.
Mistake 4
Not Applying the Coefficient
There is a common mistake that students at times forget to multiply the coefficient placed before x².
For example, they might incorrectly write d/dx (-x²) = -x, forgetting the coefficient of 2.
Students should check the coefficients in the terms and ensure they are multiplied properly.
For e.g., the correct equation is d/dx (-x²) = -2x.
Examples Using the Derivative of -x²
Problem 1
Calculate the derivative of (-x² · x³)
Here, we have f(x) = -x² · x³.
Using the product rule, f'(x) = u′v + uv′
In the given equation, u = -x² and v = x³.
Let’s differentiate each term, u′ = d/dx (-x²) = -2x v′ = d/dx (x³) = 3x² substituting into the given equation, f'(x) = (-2x)(x³) + (-x²)(3x²)
Let’s simplify terms to get the final answer, f'(x) = -2x⁴ - 3x⁴ f'(x) = -5x⁴
Thus, the derivative of the specified function is -5x⁴.
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 car's speed reduction is represented by the function y = -x², where y represents the deceleration at a distance x. If x = 3 meters, calculate the rate of change of deceleration.
We have y = -x² (deceleration of the car)...(1)
Now, we will differentiate the equation (1)
Take the derivative -x²: dy/dx = -2x
Given x = 3 (substitute this into the derivative) dy/dx = -2(3) = -6
Hence, the rate of change of deceleration at a distance x = 3 meters is -6.
Explanation
We find the rate of change of deceleration at x = 3 meters as -6, which means that at this point, the car's speed is decreasing at a rate of 6 units per meter.
Problem 3
Derive the second derivative of the function y = -x².
The first step is to find the first derivative, dy/dx = -2x...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx[-2x] d²y/dx² = -2
Therefore, the second derivative of the function y = -x² is -2.
Explanation
We use the step-by-step process, where we start with the first derivative. By differentiating -2x, we find that the second derivative is a constant value, -2.
Problem 4
Prove: d/dx (-(x²)²) = -4x³.
Let’s start using the chain rule:
Consider y = -(x²)² y = -(x⁴)
To differentiate, we use the power rule: dy/dx = -1 * 4x³ dy/dx = -4x³
Hence proved.
Explanation
In this step-by-step process, we used the power rule to differentiate the equation. We then simplify to find the derivative of the given function.
Problem 5
Solve: d/dx (-x²/x)
To differentiate the function, we use the quotient rule: d/dx (-x²/x) = (d/dx (-x²) * x - (-x²) * d/dx(x))/ x²
We will substitute d/dx (-x²) = -2x and d/dx (x) = 1 = (-2x * x - (-x²) * 1) / x² = (-2x² + x²) / x² = -x² / x² = -1
Therefore, d/dx (-x²/x) = -1
Explanation
In this process, we differentiate the given function using the quotient rule. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of -x²
1.Find the derivative of -x².
2.Can we use the derivative of -x² in real life?
3.What happens when x is 0 for the derivative of -x²?
4.What rule is used to differentiate -x²/x?
5.Are the derivatives of -x² and -2x the same?
Important Glossaries for the Derivative of -x²
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Polynomial Function: A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.
- Power Rule: A basic rule in calculus used to find the derivative of a power of x.
- First Principle: A fundamental method used to define the derivative of a function.
- Quotient Rule: A rule for finding the derivative of a quotient of two functions.
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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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