111 LearnersLast updated on August 5th, 2025
Derivative of 1/x^7
We use the derivative of 1/x^7, which is -7/x^8, to understand how this function changes with a small change in x. Derivatives are useful in calculating rates of change, like speed or growth, in real-world applications. We will now discuss the derivative of 1/x^7 in detail.
What is the Derivative of 1/x^7?
To find the derivative of 1/x^7, we represent it as d/dx (1/x^7) or (1/x^7)'. The derivative is found to be -7/x^8. This function is differentiable within its domain, indicating a smooth rate of change.
The key concepts are mentioned below:
Power Rule: A basic rule for differentiating expressions of the form x^n.
Negative Exponents: 1/x^7 can be rewritten as x^-7.
Chain Rule: Useful for functions within functions.
Derivative of 1/x^7 Formula
The derivative of 1/x^7 can be denoted as d/dx (1/x^7) or (1/x^7)'.
Using the power rule, the formula we use to differentiate 1/x^7 is: d/dx (1/x^7) = -7/x^8
This formula applies to all x where x ≠ 0.
Proofs of the Derivative of 1/x^7
We can derive the derivative of 1/x^7 through several methods. Here we demonstrate using the power rule and chain rule:
By Power Rule
Rewrite 1/x^7 as x^-7. Using the power rule, d/dx (x^n) = n*x^(n-1), we find: d/dx (x^-7) = -7*x^(-7-1) = -7/x^8.
Hence, the derivative is -7/x^8.
Using Chain Rule
Consider f(x) = 1/x^7 = (x^7)^-1. Let g(x) = x^7, then f(x) = g(x)^-1.
Using the chain rule, d/dx (g(x)^n) = n*g(x)^(n-1)*g'(x), we get: d/dx (1/x^7) = -1*(x^7)^-2*(7x^6) = -7/x^8.
Thus, the derivative is -7/x^8.
Higher-Order Derivatives of 1/x^7
When a function is differentiated multiple times, we obtain higher-order derivatives. Think of it like the acceleration of a car (second derivative) in addition to the speed (first derivative). Higher-order derivatives provide deeper insights into the behavior of functions like 1/x^7.
For the first derivative, we write f′(x), indicating the rate of change or slope at a point. The second derivative, f′′(x), is derived from the first derivative. This pattern continues for higher-order derivatives.
For the nth Derivative of 1/x^7, we denote it as f^(n)(x), showing changes in the rate of change.
Special Cases:
When x is 0, the derivative is undefined because 1/x^7 has a vertical asymptote there. When x is 1, the derivative of 1/x^7 = -7.
Common Mistakes and How to Avoid Them in Derivatives of 1/x^7
Students often make errors when differentiating 1/x^7. Understanding the correct methods can help avoid these mistakes. Here are some common errors and solutions:
Mistake 1
Ignoring Negative Exponents
Students sometimes overlook rewriting 1/x^7 as x^-7, leading to incorrect differentiation. Always rewrite the expression in terms of exponents before applying the power rule.
Mistake 2
Forgetting Undefined Points
It's easy to forget that 1/x^7 is undefined at x = 0. Always consider the domain when differentiating functions to understand where the function isn't continuous.
Mistake 3
Incorrect Use of the Power Rule
Errors occur when students misapply the power rule. For example: Incorrect differentiation: d/dx (1/x^7) = 7/x^6. Correctly apply the power rule: x^-7, so the derivative is -7/x^8.
Mistake 4
Misapplying the Chain Rule
Students sometimes misapply the chain rule when differentiating composite functions. Ensure that you identify the outer and inner functions correctly and differentiate each accurately.
Mistake 5
Missing Negative Signs
Students may forget to include the negative sign in the derivative of x^-n. For instance, write -7 instead of 7 when differentiating x^-7.
Examples Using the Derivative of 1/x^7
Problem 1
Calculate the derivative of (1/x^7)·(x^2).
Here, we have f(x) = (1/x^7)·(x^2).
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = 1/x^7 and v = x^2.
Differentiate each term: u′= d/dx (1/x^7) = -7/x^8 v′= d/dx (x^2) = 2x
Substitute into the product rule, f'(x) = (-7/x^8)·(x^2) + (1/x^7)·(2x)
Simplify to get the final answer, f'(x) = -7/x^6 + 2/x^6
Thus, the derivative of the specified function is -5/x^6.
Explanation
We find the derivative by dividing the function into two parts. First, we find their individual derivatives, then combine them using the product rule for the final result.
Problem 2
A company models a certain variable with the function y = 1/x^7. If x = 2, determine the rate of change.
We have y = 1/x^7 (the model of the variable)...(1)
Now, differentiate equation (1): dy/dx = -7/x^8
Given x = 2, substitute this into the derivative: dy/dx = -7/(2^8) dy/dx = -7/256
Hence, the rate of change when x = 2 is -7/256.
Explanation
We find the rate of change by substituting x = 2 into the derivative -7/x^8. This calculation shows the rate at which the variable changes at x = 2.
Problem 3
Derive the second derivative of the function y = 1/x^7.
First, find the first derivative: dy/dx = -7/x^8...(1)
Now, differentiate equation (1) to get the second derivative: d^2y/dx^2 = d/dx [-7/x^8]
Use the power rule: d^2y/dx^2 = 56/x^9
Therefore, the second derivative of the function y = 1/x^7 is 56/x^9.
Explanation
We use the step-by-step process, starting with the first derivative. Using the power rule again, we differentiate -7/x^8 to find the second derivative, 56/x^9.
Problem 4
Prove: d/dx (x^2/x^7) = -5/x^6.
Rewrite x^2/x^7 as x^-5.
Differentiate using the power rule: d/dx (x^-5) = -5*x^(-5-1) d/dx (x^-5) = -5/x^6
Hence proved.
Explanation
In this step-by-step process, we rewrite the function with a negative exponent, differentiate using the power rule, and simplify to derive the equation.
Problem 5
Solve: d/dx (x/x^7).
Rewrite x/x^7 as x^-6.
Differentiate using the power rule: d/dx (x^-6) = -6*x^(-6-1) d/dx (x^-6) = -6/x^7
Therefore, d/dx (x/x^7) = -6/x^7.
Explanation
In this process, we rewrite the given function with a negative exponent and differentiate using the power rule to simplify the equation and find the final result.
FAQs on the Derivative of 1/x^7
1.Find the derivative of 1/x^7.
2.Can we use the derivative of 1/x^7 in real life?
3.Is it possible to take the derivative of 1/x^7 at the point where x = 0?
4.What rule is used to differentiate 1/x^7?
5.Are the derivatives of 1/x^7 and (1/x)^7 the same?
Important Glossaries for the Derivative of 1/x^7
- Derivative: The derivative of a function indicates how the 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: Represents the reciprocal of a base raised to a positive exponent, e.g., x^-n = 1/x^n.
- Chain Rule: A rule used to differentiate composite functions.
- Asymptote: A line that a graph approaches but never touches or crosses.
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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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