Summarize this article:
101 LearnersLast updated on September 15, 2025
Derivative of -x/y
We use the derivative of -x/y to understand how this function changes with respect to a slight change in x or y. Derivatives are essential tools in calculus that help us solve problems in various fields, including physics and economics. We will now discuss the derivative of -x/y in detail.
What is the Derivative of -x/y?
We now explore the derivative of -x/y. It is represented as d/dx (-x/y) or (-x/y)'. The derivative of -x/y is determined using the quotient rule, as it is a ratio of functions. This derivative indicates how the function changes within its domain. The key concepts are as follows:
Function Representation: (-x/y) = -1 * (x/y).
Quotient Rule: Rule for differentiating -x/y.
Negative Constant: Consideration of the negative constant factor in differentiation.
Derivative of -x/y Formula
The derivative of -x/y with respect to x can be denoted as d/dx (-x/y) or (-x/y)'.
The formula we use to differentiate -x/y is: d/dx (-x/y) = (-1/y) * (y - x(dy/dx))/y²
This applies to all values where y ≠ 0.
Proofs of the Derivative of -x/y
We derive the derivative of -x/y using proofs. This involves applying the quotient rule and understanding the role of constants. Here are the methods used to prove this:
- Using the Quotient Rule
- Considering Constant Multiplication
We will now demonstrate that the differentiation of -x/y results in the formula mentioned above using these methods:
Using the Quotient Rule
To find the derivative of -x/y using the quotient rule, consider the function f(x) = -x/y.
The derivative can be expressed as:
f'(x) = d/dx (-x/y) = d/dx (-1 * (x/y)) = -1 * [ (y * d/dx(x) - x * d/dx(y)) / y² ] = -1 * [ (y - x(dy/dx)) / y² ] = (-1/y) * (y - x(dy/dx))/y²
This result demonstrates the application of the quotient rule and the effect of the negative constant.
Higher-Order Derivatives of -x/y
When a function is differentiated multiple times, the results are known as higher-order derivatives. These can become more complex as the order increases. Consider the analogy of a car where speed (first derivative) changes, and the rate of this change (second derivative) also varies. Higher-order derivatives aid in understanding functions like -x/y.
For the first derivative of a function, we write f′(x), indicating how the function changes or its slope at a point. The second derivative, f′′(x), is derived from the first derivative, and this pattern continues for higher orders.
For the nth derivative of -x/y, we use f⁽ⁿ⁾(x) to represent the change in the rate of change.
Special Cases:
At points where y = 0, the derivative is undefined because -x/y is not defined there. When x = 0, the derivative of -x/y depends on the value of y, and it simplifies to 0 when y is constant.
Common Mistakes and How to Avoid Them in Derivatives of -x/y
Students often make mistakes when differentiating -x/y. These errors can be resolved by understanding the correct methods. Here are some common mistakes and ways to address them:
Mistake 1
Ignoring the Negative Sign
Students may overlook the negative sign, leading to incorrect results. The negative sign is crucial in the differentiation process, as it affects the entire derivative. Ensure to consider the negative constant throughout the calculation.
Mistake 2
Forgetting to Apply the Quotient Rule
Some may forget to apply the quotient rule when differentiating -x/y, leading to errors. Remember that -x/y is a ratio, and the quotient rule is essential. Properly apply the quotient rule to avoid mistakes.
Mistake 3
Misapplying the Quotient Rule
Misapplication of the quotient rule can occur, especially when dealing with complex fractions. For example, incorrectly differentiating: d/dx (-x/y) = y/x². Correct differentiation: d/dx (-x/y) = (-y - x(dy/dx))/y². To avoid this, practice applying the quotient rule accurately.
Mistake 4
Not Considering Undefined Points
Some students might not consider that the function -x/y is undefined when y = 0. It is crucial to recognize the domain of the function and understand discontinuities to avoid errors.
Mistake 5
Ignoring Constants and Coefficients
There is a common mistake where students forget to multiply constants before differentiating. For example, ignoring the negative sign in d/dx (-5x/y). Ensure constants are correctly applied in the derivative process. The correct equation is d/dx (-5x/y) = -5(y - x(dy/dx))/y².
Examples Using the Derivative of -x/y
Problem 1
Calculate the derivative of (-2x/y).
Here, we have f(x) = -2x/y.
Using the quotient rule, f'(x) = d/dx (-2x/y) = -2 * [ (y - x(dy/dx)) / y² ] = -2(y - x(dy/dx))/y²
Thus, the derivative of the specified function is -2(y - x(dy/dx))/y².
Explanation
We find the derivative of the given function by applying the quotient rule, considering the negative constant factor, and simplifying the expression to obtain the result.
Problem 2
A water tank is being drained at a rate represented by the function h = -x/y, where h is the height of the water and x is the time. If y = 5 seconds, find the rate of change of the water height when x = 10 seconds.
We have h = -x/y (rate of change of water height)...(1)
Now, we will differentiate the equation (1) with respect to x. dh/dx = d/dx (-x/5) = -1/5 * (5 - x(dy/dx))/5² = -1/5
Hence, the rate of change of the water height at x = 10 seconds is -1/5.
Explanation
We find the rate of change of the water height by differentiating the function with respect to time and substituting the given values into the derivative.
Problem 3
Derive the second derivative of the function h = -x/y.
The first step is to find the first derivative, dh/dx = (-y - x(dy/dx))/y²...(1)
Now, we will differentiate equation (1) to get the second derivative: d²h/dx² = d/dx [(-y - x(dy/dx))/y²] = d/dx (-1/y² * (y - x(dy/dx))) = (2/y³) * (y - x(dy/dx))
Therefore, the second derivative of the function h = -x/y is (2/y³) * (y - x(dy/dx)).
Explanation
We use a step-by-step process, starting with the first derivative.
Using the quotient rule, we differentiate further to find the second derivative, considering the constant factors.
Problem 4
Prove: d/dx (-x²/y) = (-2x/y) - (x²/y²)(dy/dx).
Let's start by using the quotient rule: Consider h = -x²/y
To differentiate, we apply the quotient rule: dh/dx = d/dx (-x²/y) = (-y * 2x - x² * dy/dx)/y² = (-2xy - x²(dy/dx))/y²
Therefore, d/dx (-x²/y) = (-2x/y) - (x²/y²)(dy/dx), hence proved.
Explanation
In this step-by-step process, we apply the quotient rule to differentiate the equation, substitute values, and simplify to derive the equation as required.
Problem 5
Solve: d/dx (-3x/y).
To differentiate the function, we use the quotient rule: d/dx (-3x/y) = -3 * [(y - x(dy/dx))/y²] = -3(y - x(dy/dx))/y² Therefore, d/dx (-3x/y) = -3(y - x(dy/dx))/y².
Explanation
In this process, we differentiate the given function using the quotient rule, accounting for the negative constant, and simplify the expression to obtain the final result.
FAQs on the Derivative of -x/y
1.Find the derivative of -x/y.
2.Can we use the derivative of -x/y in real life?
3.Is it possible to take the derivative of -x/y when y = 0?
4.What rule is used to differentiate -x²/y?
5.Does the derivative of -x/y differ from that of a positive x/y?
Important Glossaries for the Derivative of -x/y
- Derivative: The derivative of a function measures how the function changes concerning a change in its variables.
- Quotient Rule: A rule used for differentiating a ratio of two functions.
- Constant Factor: A fixed value that affects the differentiation of a function when multiplied by it.
- Undefined: A term used when a function or derivative does not exist at a certain point or value.
- Higher-Order Derivative: The result of differentiating a function multiple times, providing insights into its behavior and changes.
Explore More calculus
![Important Math Links Icon]()
Previous to Derivative of -x/y
![Important Math Links Icon]()
Next to Derivative of -x/y
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.
