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119 LearnersLast updated on 20 September 2025
Derivative of y with respect to x
We use the derivative of y with respect to x as a measuring tool for how a function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now discuss the derivative of y with respect to x in detail.
What is the Derivative of y with respect to x?
We now understand the derivative of y with respect to x. It is commonly represented as dy/dx and indicates how the function y changes as x changes. If a function y is differentiable within its domain, it has a clearly defined derivative. The key concepts are mentioned below:
Function: The relationship between x and y, where y is expressed as a function of x.
Derivative: The measure of how a function changes as its input changes.
Differentiability: A property indicating that a function has a derivative at every point in its domain.
Derivative of y with respect to x Formula
The derivative of y with respect to x can be denoted as dy/dx. If y is a function of x, the formula used to differentiate y with respect to x is based on the specific form of y.
The formula applies to all x where the function y is continuous and differentiable.
Proofs of the Derivative of y with respect to x
We can derive the derivative of y with respect to x using various proofs. To show this, we will use mathematical principles and rules of differentiation. There are several methods we use, such as:
- By First Principle
- Using Chain Rule
- Using Product Rule
We will now demonstrate the differentiation of y with respect to x using the above-mentioned methods:
By First Principle
The derivative of y with respect to x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.
To find the derivative of y using the first principle, we will consider f(x) = y. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h ... (1)
Substituting y into equation (1), f'(x) = limₕ→₀ [y(x + h) - y(x)] / h = limₕ→₀ Δy / h where Δy = y(x + h) - y(x).
Using Chain Rule
To prove the differentiation of y using the chain rule, Consider y as a composite function. By chain rule, if y = g(f(x)), the derivative is dy/dx = g'(f(x))f'(x).
Using Product Rule
We will now prove the derivative of a product of functions using the product rule. If y = u(x)v(x), then by product rule: dy/dx = u'(x)v(x) + u(x)v'(x).
Higher-Order Derivatives of y with respect to x
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 y(x).
For the first derivative of a function, we write dy/dx, 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 d²y/dx². Similarly, the third derivative, d³y/dx³ is the result of the second derivative and this pattern continues.
For the nth derivative of y(x), we generally use dⁿy/dxⁿ, which tells us the change in the rate of change.
Special Cases:
When the function y has a vertical asymptote at a certain point, the derivative is undefined at that point. When y is a constant function, the derivative dy/dx = 0.
Common Mistakes and How to Avoid Them in Derivatives of y with respect to x
Students frequently make mistakes when differentiating functions. 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 the product or chain rule. 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 the Undefined Points
They might not remember that some functions are undefined at certain points. 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 certain points.
Mistake 3
Incorrect use of Differentiation Rules
While differentiating functions, students might misapply differentiation rules. For example: Incorrect differentiation of a quotient. Applying the quotient rule, d/dx (u/v) = (v . u' - u . v')/ v². To avoid this mistake, write the quotient rule without errors. 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 a function. For example, they incorrectly write d/dx (5y) = dy/dx. Students should check the constants in the terms and ensure they are multiplied properly. For example, the correct equation is d/dx (5y) = 5(dy/dx).
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 (f(g(x))) = f'(g(x)). 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 (f(g(x))) = f'(g(x))g'(x).
Examples Using the Derivative of y with respect to x
Problem 1
Calculate the derivative of (y·y²)
Here, we have f(x) = y·y². Using the product rule, f'(x) = u′v + uv′ In the given equation, u = y and v = y².
Let’s differentiate each term, u′= dy/dx v′= d/dx (y²) = 2y(dy/dx)
substituting into the given equation, f'(x) = (dy/dx)·y² + y·2y(dy/dx)
Let’s simplify terms to get the final answer, f'(x) = y²(dy/dx) + 2y²(dy/dx).
Thus, the derivative of the specified function is 3y²(dy/dx).
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 water tank is being filled with water at a rate that causes its height to change according to the function y = x², where y represents the height of the water in meters, and x is time in seconds. If x = 2 seconds, calculate the rate of change of the height of the water.
We have y = x² (the height of the water)...(1)
Now, we will differentiate the equation (1) Take the derivative of x²: dy/dx = 2x Given x = 2 (substitute this into the derivative) dy/dx = 2(2) = 4.
Hence, the rate of change of the height of the water at x = 2 seconds is 4 meters/second.
Explanation
We find the rate of change of the height of the water at x = 2 seconds as 4 meters/second, which means that at this time, the height of the water increases at a rate of 4 meters per second.
Problem 3
Derive the second derivative of the function y = x³.
The first step is to find the first derivative, dy/dx = 3x²...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [3x²] d²y/dx² = 6x.
Therefore, the second derivative of the function y = x³ is 6x.
Explanation
We use the step-by-step process, where we start with the first derivative. By differentiating it, we find the second derivative, which is 6x.
Problem 4
Prove: d/dx (x²) = 2x.
Consider y = x². To differentiate, we use the power rule: dy/dx = 2x. Hence proved.
Explanation
In this step-by-step process, we used the power rule to differentiate the equation. We directly obtain the derivative as 2x.
Problem 5
Solve: d/dx (y/x)
To differentiate the function, we use the quotient rule: d/dx (y/x) = (d/dx (y)·x - y·d/dx(x))/x²
We will substitute d/dx (y) = dy/dx and d/dx (x) = 1 = (x(dy/dx) - y)/x².
Therefore, d/dx (y/x) = (x(dy/dx) - y)/x².
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 y with respect to x
1.Find the derivative of y = x².
2.Can we use the derivative of y with respect to x in real life?
3.Is it possible to take the derivative of y at a point where the function is not defined?
4.What rule is used to differentiate y/x?
5.Are the derivatives of y and y⁻¹ the same?
Important Glossaries for the Derivative of y with respect to x
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Continuous Function: A function that is continuous at every point in its domain.
- Quotient Rule: A rule used to differentiate expressions in the form of a quotient.
- Chain Rule: A rule used to differentiate composite functions.
- Higher-Order Derivative: Derivatives obtained by differentiating a function multiple times.
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.
