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109 LearnersLast updated on September 22, 2025
Derivative of x^-3
We use the derivative of x^-3, which is -3x^-4, to understand how the function changes in response to a slight change in x. Derivatives play a crucial role in various real-life applications, such as calculating rates of change. We will now discuss the derivative of x^-3 in detail.
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 -3x^-4. The function x^-3 has a well-defined derivative, indicating it is differentiable across its domain.
The key concepts are mentioned below:
Power Rule: A fundamental rule for differentiating functions of the form x^n.
Negative Exponents: Understanding how to differentiate functions with negative powers.
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) = -3x^-4
The formula applies to all x except where x = 0, as division by zero is undefined.
Proofs of the Derivative of x^-3
We can derive the derivative of x^-3 using proofs. To show this, we will use differentiation rules. There are several methods we use to prove this, such as:
- By Power Rule
- By First Principle
By Power Rule
The derivative of x^-3 can be easily found using the power rule, which states that d/dx (x^n) = n*x^(n-1). For x^-3, n = -3. Thus, d/dx (x^-3) = -3*x^(-3-1) = -3x^-4.
By First Principle
The derivative of x^-3 can also 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 consider f(x) = x^-3. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h
Given that f(x) = x^-3, we write f(x + h) = (x + h)^-3.
Substituting these into the equation, f'(x) = limₕ→₀ [(x + h)^-3 - x^-3] / h
Simplifying using binomial expansion and limits, f'(x) = -3x^-4.
Hence, proved.
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 offer insights into the behavior of functions.
For the first derivative of a function, we write f′(x), indicating how the function changes or its slope at a particular point. The second derivative is derived from the first derivative, 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, continuing for higher-order derivatives.
Special Cases:
When x = 0, the derivative is undefined because x^-3 is not defined at zero. When x = 1, the derivative of x^-3 = -3*1^-4 = -3.
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
Not applying the power rule correctly
Students may forget to apply the power rule correctly, leading to incomplete or incorrect results. They often skip steps and directly arrive at the result. Ensure that each step is followed systematically to avoid errors in the process.
Mistake 2
Misinterpreting negative exponents
Students might misinterpret negative exponents, leading to confusion in differentiation. Remember that a negative exponent means the reciprocal of the positive power of the base.
Mistake 3
Ignoring undefined points
They might not remember that x^-3 is undefined at x = 0. Keep in mind that division by zero is undefined, so the function is not continuous at x = 0.
Mistake 4
Incorrect simplification
There is a common mistake where students incorrectly simplify expressions with negative exponents. Ensure proper application of algebraic rules when simplifying such expressions.
Mistake 5
Not considering higher-order derivatives
Students often stop at the first derivative without considering higher-order derivatives, which can provide valuable insights into the behavior of the function.
Examples Using the Derivative of x^-3
Problem 1
Calculate the derivative of (x^-3 * x^2)
Here, we have f(x) = x^-3 * x^2.
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = x^-3 and v = x^2.
Let’s differentiate each term, u′= d/dx (x^-3) = -3x^-4 v′= d/dx (x^2) = 2x
Substituting into the given equation, f'(x) = (-3x^-4) * (x^2) + (x^-3) * (2x) = -3x^-2 + 2x^-2
Simplifying terms to get the final answer, f'(x) = -x^-2
Thus, the derivative of the specified function is -x^-2.
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’s depreciation of a machine is modeled by the function y = x^-3, where y represents the depreciation value at time x. If x = 2 years, calculate the rate of depreciation.
We have y = x^-3 (depreciation model)...(1)
Now, we will differentiate the equation (1)
Take the derivative x^-3: dy/dx = -3x^-4 Given x = 2 (substitute this into the derivative) dy/dx = -3*(2^-4) = -3/16
Hence, we get the rate of depreciation at a time x = 2 years as -3/16.
Explanation
We find the rate of depreciation at x = 2 as -3/16, which indicates that the depreciation value decreases at this rate at the given point in time.
Problem 3
Derive the second derivative of the function y = x^-3.
The first step is to find the first derivative, dy/dx = -3x^-4...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-3x^-4] d²y/dx² = 12x^-5
Therefore, the second derivative of the function y = x^-3 is 12x^-5.
Explanation
We use the step-by-step process, where we start with the first derivative. Using the power rule, we differentiate -3x^-4 to find the second derivative: 12x^-5.
Problem 4
Prove: d/dx ((x^-3)^2) = -6x^-7.
Let’s start using the chain rule: Consider y = (x^-3)^2 = (x^-3)^2
To differentiate, we use the chain rule: dy/dx = 2(x^-3) * d/dx [x^-3]
Since the derivative of x^-3 is -3x^-4, dy/dx = 2(x^-3) * (-3x^-4) = -6x^-7
Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace x^-3 with its derivative. As a final step, we simplify to derive the equation.
Problem 5
Solve: d/dx (x^-3/x)
To differentiate the function, we use the quotient rule: d/dx (x^-3/x) = (d/dx (x^-3) * x - x^-3 * d/dx(x))/x²
We will substitute d/dx (x^-3) = -3x^-4 and d/dx (x) = 1 = ( (-3x^-4) * x - x^-3 * 1) / x² = (-3x^-3 - x^-3) / x² = -4x^-3 / x² = -4x^-5
Therefore, d/dx (x^-3/x) = -4x^-5.
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 x^-3
1.Find the derivative of x^-3.
2.Can we use the derivative of x^-3 in real life?
3.Is it possible to take the derivative of x^-3 at the point where x = 0?
4.What rule is used to differentiate x^-3/x?
5.Are the derivatives of x^-3 and (x^3)^-1 the same?
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.
- Power Rule: A fundamental rule used to differentiate functions of the form x^n.
- Negative Exponent: A concept in mathematics where a negative exponent indicates the reciprocal of the positive exponent.
- First Derivative: The initial result of differentiating a function, giving the rate of change of the function.
- Undefined Point: A point at which a function is not defined, often due to division by zero or other mathematical restrictions.
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Jaskaran Singh Saluja
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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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