100 LearnersLast updated on July 21st, 2025
Derivative of y
We use the derivative of y, which measures how the function y changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now talk about the derivative of y in detail.
What is the Derivative of y?
We now understand the derivative of y. It is commonly represented as dy/dx or y', and its value depends on the specific function y represents.
The function y has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
- Function y: A general function that can be any mathematical expression.
- Differentiation: The process of finding the derivative of a function.
- Rate of Change: The speed at which a variable changes over a specific period.
Derivative of y Formula
The derivative of y can be denoted as dy/dx or y'. The formula we use to differentiate y depends on the specific form of the function y.
The general derivative formula applies to all x where the function y is defined and differentiable.
Proofs of the Derivative of y
We can derive the derivative of y using proofs. To show this, we will use mathematical identities along with the rules of differentiation.
There are several methods we use to prove this, such as: - By First Principle - Using Chain Rule - Using Product
Rule We will now demonstrate the differentiation of y using the above-mentioned methods: By First Principle The derivative of y 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, consider f(x) = y. Its derivative can be expressed as the following limit: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Using Chain Rule To prove the differentiation of y using the chain rule, we consider y as a composition of two or more functions.
If y = g(f(x)), then the chain rule states: dy/dx = g'(f(x)) * f'(x) Using Product Rule We will now prove the derivative of y using the product rule. If y = u(x) * v(x), the product rule states: dy/dx = u'(x) * v(x) + u(x) * v'(x)
Higher-Order Derivatives of y
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. 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 y, 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.
Special Cases:
When the function y has points where it is undefined, the derivative is also undefined at those points.
When y is a constant function, the derivative of y is zero because a constant does not change with x.
Common Mistakes and How to Avoid Them in Derivatives of y
Students frequently make mistakes when differentiating y.
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 of y
They might not remember that y might be 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 such certain points.
Mistake 3
Incorrect use of Differentiation Rules
While differentiating complex functions, students may misapply differentiation rules.
For example, incorrect differentiation of a product or quotient of two functions.
To avoid this mistake, write the differentiation rules 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 sometimes forget to multiply the constants placed before y.
For example, they might incorrectly write dy/dx = 1 instead of taking into account any coefficients.
Students should check the constants in the terms and ensure they are multiplied properly.
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.
To fix this error, students should divide the functions into inner and outer parts.
Then, make sure that each function is differentiated.
Examples Using the Derivative of y
Problem 1
Calculate the derivative of (y * e^x)
Here, we have f(x) = y * e^x. Using the product rule, f'(x) = u′v + uv′ In the given equation, u = y and v = e^x. Let’s differentiate each term, u′= dy/dx (y) v′= d/dx (e^x) = e^x
Substituting into the given equation, f'(x) = (dy/dx) * e^x + y * e^x Thus, the derivative of the specified function is (dy/dx) * e^x + y * e^x.
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 observes that the cost of production is given by the function y = 3x^2. If x = 4 units, measure the cost change with respect to production.
We have y = 3x^2 (cost of production)...(1) Now, we will differentiate the equation (1) Take the derivative of y: dy/dx = 6x Given x = 4 (substitute this into the derivative) dy/dx = 6 * 4 = 24
Hence, the rate of change of cost with respect to production at x = 4 units is 24.
Explanation
We find the rate of change of cost at x = 4 units as 24, which means that at this point, the cost increases at a rate of 24 units per unit increase in production.
Problem 3
Derive the second derivative of the function y = x^3.
The first step is to find the first derivative, dy/dx = 3x^2...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [3x²] = 6x
Therefore, the second derivative of the function y = x^3 is 6x.
Explanation
We use the step-by-step process, where we start with the first derivative. We then differentiate the first derivative to find the second derivative.
Problem 4
Prove: d/dx (x²y) = 2xy + x²(dy/dx).
Let’s start using the product rule: Consider y as a function of x, and let f(x) = x²y.
To differentiate, we use the product rule: d/dx (x²y) = d/dx (x²) * y + x² * d/dx (y) = 2xy + x²(dy/dx) Hence proved.
Explanation
In this step-by-step process, we used the product rule to differentiate the equation.
We differentiate both terms separately and combine them to get the result.
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 (x) = 1 = (x * dy/dx - y * 1) / x² = (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
1.Find the derivative of y.
2.Can we use the derivative of y in real life?
3.Is it possible to take the derivative of y at a point where y is undefined?
4.What rule is used to differentiate y/x?
5.Does the derivative of y depend on the form of y?
Important Glossaries for the Derivative of y
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Function: A mathematical expression involving one or more variables.
- Product Rule: A rule used in calculus to find the derivative of a product of two functions.
- Quotient Rule: A rule used in calculus to find the derivative of a quotient of two functions.
- Chain Rule: A rule in calculus used to differentiate the composition of two or more 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.
