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105 LearnersLast updated on September 12, 2025
Derivative of Negative Exponent
We use the derivative of a negative exponent function, which is a key concept in calculus, to understand how functions with negative powers change in response to slight changes in x. Derivatives are vital for calculating profit or loss and understanding rates of change in real-life situations. We will now discuss the derivative of negative exponent functions in detail.
What is the Derivative of a Negative Exponent Function?
We now understand the derivative of a function with a negative exponent. It is commonly represented as d/dx (x⁻ⁿ) or (x⁻ⁿ)', and its value is -n·x⁻ⁿ⁻¹.
Functions with negative exponents have clearly defined derivatives, indicating they are differentiable within their domain. The key concepts are mentioned below:
Power Rule: Rule for differentiating functions of the form xⁿ, applicable to x⁻ⁿ as well.
Negative Exponent: x⁻ⁿ = 1/xⁿ, an important identity to understand.
Derivative of a Negative Exponent Formula
The derivative of x⁻ⁿ can be denoted as d/dx (x⁻ⁿ) or (x⁻ⁿ)'.
The formula we use to differentiate x⁻ⁿ is: d/dx (x⁻ⁿ) = -n·x⁻ⁿ⁻¹ (or) (x⁻ⁿ)' = -n·x⁻ⁿ⁻¹
The formula applies to all x where x ≠ 0.
Proofs of the Derivative of a Negative Exponent
We can derive the derivative of x⁻ⁿ 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 Principle
- Using Power Rule
- Using Negative Exponent
Identity We will now demonstrate that the differentiation of x⁻ⁿ results in -n·x⁻ⁿ⁻¹ using the above-mentioned methods:
By First Principle
The derivative of x⁻ⁿ can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.
To find the derivative of x⁻ⁿ using the first principle, we will consider f(x) = x⁻ⁿ. 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⁻ⁿ, we write f(x + h) = (x + h)⁻ⁿ.
Substituting these into equation (1), f'(x) = limₕ→₀ [(x + h)⁻ⁿ - x⁻ⁿ] / h
Using binomial expansion and simplifying, f'(x) = -n·x⁻ⁿ⁻¹ Hence, proved.
Using Power Rule
To prove the differentiation of x⁻ⁿ using the power rule, We use the formula: d/dx (xⁿ) = n·xⁿ⁻¹ Set n to be negative, i.e., n = -m, d/dx (x⁻ⁿ) = -n·x⁻ⁿ⁻¹
Thus, the derivative of x⁻ⁿ is -n·x⁻ⁿ⁻¹.
Using Negative Exponent
Identity Consider x⁻ⁿ = 1/xⁿ Differentiate using the quotient rule, d/dx (1/xⁿ) = -n·x⁻ⁿ⁻¹
Hence, the derivative of x⁻ⁿ is -n·x⁻ⁿ⁻¹.
Higher-Order Derivatives of Negative Exponent Functions
When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a bit challenging.
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⁻ⁿ.
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⁻ⁿ, 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 is undefined because x⁻ⁿ involves division by zero at this point.
When n = 1, the derivative of x⁻¹ = -1·x⁻², which is -1/x².
Common Mistakes and How to Avoid Them in Derivatives of Negative Exponents
Students frequently make mistakes when differentiating functions with negative exponents. 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 expression
Students may forget to simplify the expression, which can lead to incomplete or incorrect results. They often skip steps and directly arrive at the result, especially when solving using the power rule. Ensure that each step is written in order. Students might think it is cumbersome, but it is important to avoid errors in the process.
Mistake 2
Forgetting the Undefined Points
They might not remember that x⁻ⁿ 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 x⁻ⁿ, students misapply the power rule. For example: Incorrect differentiation: d/dx (x⁻ⁿ) = nx⁻ⁿ⁻¹. The correct differentiation: d/dx (x⁻ⁿ) = -n·x⁻ⁿ⁻¹. To avoid this mistake, write the power rule correctly. Always check for errors in the calculation and ensure it is properly simplified.
Mistake 4
Not writing Constants and Coefficients
There is a common mistake that students at times forget to multiply the constants placed before x⁻ⁿ. For example, they incorrectly write d/dx (5x⁻ⁿ) = x⁻ⁿ⁻¹. Students should check the constants in the terms and ensure they are multiplied properly. For example, the correct equation is d/dx (5x⁻ⁿ) = -5n·x⁻ⁿ⁻¹.
Mistake 5
Not Applying the Chain Rule
Students often forget to use the chain rule. This happens when the derivative of the inner function is not considered. For example: Incorrect: d/dx ((x + 1)⁻²) = -2(x + 1)⁻³. To fix this error, students should divide the functions into inner and outer parts. Then, make sure that each function is differentiated. For example, d/dx ((x + 1)⁻²) = -2(x + 1)⁻³·1.
Examples Using the Derivative of Negative Exponents
Problem 1
Calculate the derivative of (x⁻² · x⁻³)
Here, we have f(x) = x⁻² · x⁻³. Using the product rule, f'(x) = u′v + uv′ In the given equation, u = x⁻² and v = x⁻³. Let’s differentiate each term, u′ = d/dx (x⁻²) = -2x⁻³ v′ = d/dx (x⁻³) = -3x⁻⁴
Substituting into the given equation, f'(x) = (-2x⁻³)(x⁻³) + (x⁻²)(-3x⁻⁴)
Let’s simplify terms to get the final answer, f'(x) = -2x⁻⁶ - 3x⁻⁶ f'(x) = -5x⁻⁶
Thus, the derivative of the specified function is -5x⁻⁶.
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 lamp post casts a shadow represented by the function y = x⁻¹, where y represents the length of the shadow at a distance x from the lamp post. If x = 2 meters, measure the rate of change of the shadow length.
We have y = x⁻¹ (length of the shadow)...(1)
Now, we will differentiate the equation (1)
Take the derivative x⁻¹: dy/dx = -1·x⁻² Given x = 2 (substitute this into the derivative)
dy/dx = -1·2⁻² dy/dx = -1/4
Hence, we get the rate of change of the shadow length at a distance x = 2 as -1/4.
Explanation
We find the rate of change of the shadow length at x = 2 as -1/4, which means that at a given point, the length of the shadow decreases at a rate of 1/4 units per unit increase in distance from the lamp post.
Problem 3
Derive the second derivative of the function y = x⁻².
The first step is to find the first derivative, dy/dx = -2x⁻³...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-2x⁻³]
Here we use the power rule, d²y/dx² = -2(-3)x⁻⁴ d²y/dx² = 6x⁻⁴
Therefore, the second derivative of the function y = x⁻² is 6x⁻⁴.
Explanation
We use the step-by-step process, where we start with the first derivative. Using the power rule, we differentiate -2x⁻³. We then simplify the terms to find the final answer.
Problem 4
Prove: d/dx ((x⁻¹)²) = -2x⁻³.
Let’s start using the chain rule: Consider y = (x⁻¹)²
To differentiate, we use the chain rule: dy/dx = 2(x⁻¹)·d/dx [x⁻¹]
Since the derivative of x⁻¹ is -1·x⁻², dy/dx = 2(x⁻¹)(-1·x⁻²)
dy/dx = -2x⁻³ Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace x⁻¹ with its derivative. As a final step, we simplify to derive the equation.
Problem 5
Solve: d/dx (x⁻²/x)
To differentiate the function, we use the quotient rule: d/dx (x⁻²/x) = (d/dx (x⁻²)·x - x⁻²·d/dx(x))/x²
We will substitute d/dx (x⁻²) = -2x⁻³ and d/dx(x) = 1 = (-2x⁻³·x - x⁻²·1)/x² = (-2x⁻² - x⁻²)/x² = -3x⁻²/x² = -3x⁻⁴
Therefore, d/dx (x⁻²/x) = -3x⁻⁴.
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 Negative Exponents
1.Find the derivative of x⁻².
2.Can we use the derivative of negative exponents in real life?
3.Is it possible to take the derivative of x⁻ⁿ at the point where x = 0?
4.What rule is used to differentiate x⁻²/x?
5.Are the derivatives of x⁻² and x² the same?
Important Glossaries for the Derivative of Negative Exponents
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Negative Exponent: A term of the form x⁻ⁿ, which is equal to 1/xⁿ.
- Power Rule: A basic rule in calculus used to differentiate functions of the form xⁿ.
- Chain Rule: A rule used to differentiate compositions of functions.
- Quotient Rule: A rule used to differentiate functions that are ratios of other functions.
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.
