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116 LearnersLast updated on September 9, 2025
Derivative of y²
We use the derivative of y², which is 2y(dy/dx), as a tool to understand how the function changes in response to a small change in x. Derivatives are crucial in calculating rates of change in various contexts. We will now discuss the derivative of y² in detail.
What is the Derivative of y²?
We now understand the derivative of y².
It is commonly represented as d/dx (y²) or (y²)', and its value is 2y(dy/dx).
The function y² has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Square Function: (y²) represents the square of y.
Chain Rule: Rule for differentiating y² (since it involves y, which is a function of x).
Differentiation: Process of finding the derivative of a function.
Derivative of y² Formula
The derivative of y² can be denoted as d/dx (y²) or (y²)'. The formula we use to differentiate y² is: d/dx (y²) = 2y(dy/dx) The formula applies to all x where y is differentiable.
Proofs of the Derivative of y²
We can derive the derivative of y² using proofs.
To show this, we will use the chain rule of differentiation.
The method we use is as follows:
Using Chain Rule
To prove the differentiation of y² using the chain rule, we use the formula:
Let y = f(x) be a function of x.
Then y² = [f(x)]². Using the chain rule: d/dx [y²] = 2y(dy/dx)
We differentiate the outer function and multiply by the derivative of the inner function.
Let's demonstrate that the differentiation of y² results in 2y(dy/dx) using this method:
Consider y² = (y)², where y = f(x). d/dx (y²) = 2y(dy/dx)
The derivative of y² with respect to x is derived by using the chain rule, multiplying the derivative of the outer function 2y by the derivative of the inner function (dy/dx), which results in 2y(dy/dx).
Hence, proved.
Higher-Order Derivatives of y²
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, consider an object 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 y².
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 y², we generally use fⁿ(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 y = 0, the derivative is 0 because any constant multiple of 0 is 0. When y is a constant, the derivative of y² = 0, since the derivative of a constant is 0.
Common Mistakes and How to Avoid Them in Derivatives of y²
Students frequently make mistakes when differentiating y². 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 Chain Rule
Students may forget to use the chain rule, which leads to incorrect results. They often skip the step of multiplying by (dy/dx) when differentiating y². Ensure that the chain rule is correctly applied, and (dy/dx) is included in the final result.
Mistake 2
Ignoring the Variable Nature of y
They might mistakenly treat y as a constant instead of a variable dependent on x. Remember that y is a function of x, so its derivative must be considered. Always apply the chain rule to account for dy/dx.
Mistake 3
Incorrect use of Power Rule
While differentiating y², students might misapply the power rule.
For example: Incorrect differentiation: d/dx (y²) = 2y.
Correct differentiation: d/dx (y²) = 2y(dy/dx). To avoid this mistake, remember to multiply by dy/dx.
Mistake 4
Overlooking Constant Multipliers
Students sometimes forget to multiply by constants present in terms involving y².
For example, they might incorrectly write d/dx (3y²) = 2y(dy/dx).
Check for constants and ensure they are multiplied properly.
For e.g., the correct equation is d/dx (3y²) = 6y(dy/dx).
Mistake 5
Missing Higher Derivatives
In problems requiring higher derivatives, students might stop after finding the first derivative.
For example, when asked for the second derivative, they might only find the first derivative.
Continue differentiating until the required order is reached.
Examples Using the Derivative of y²
Problem 1
Calculate the derivative of (y² · sin y).
Here, we have f(x) = y² · sin y.
Using the product rule, f'(x) = u′v + uv′
In the given equation, u = y² and v = sin y.
Let’s differentiate each term, u′ = d/dx (y²) = 2y(dy/dx) v′ = d/dx (sin y) = cos y(dy/dx)
Substituting into the given equation, f'(x) = (2y(dy/dx)) · (sin y) + (y²) · (cos y(dy/dx))
Let’s simplify terms to get the final answer, f'(x) = 2y sin y(dy/dx) + y² cos y(dy/dx)
Thus, the derivative of the specified function is 2y sin y(dy/dx) + y² cos y(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 balloon rises such that its height is represented by the function y² = 4x. If x = 9 meters, find the rate of change of the balloon's height.
We have y² = 4x (height of the balloon)...(1)
Now, we will differentiate the equation (1)
Take the derivative of y²: d/dx (y²) = 2y(dy/dx) = 4
Given x = 9, we can find y: y² = 4(9) y = 6 (assuming y is positive)
Substitute y = 6 into the derivative: 2(6)(dy/dx) = 4 12(dy/dx) = 4 dy/dx = 4/12 = 1/3
Hence, the rate of change of the balloon's height at x = 9 meters is 1/3.
Explanation
We find the rate of change by differentiating the function representing the balloon's height. We then solve for dy/dx using the given x value and simplify to find the final rate.
Problem 3
Derive the second derivative of the function y² = x².
The first step is to find the first derivative, d/dx (y²) = 2y(dy/dx) = 2x...(1)
Now we will differentiate equation (1) to get the second derivative: d²/dx² (2y(dy/dx)) = d/dx (2x) d²/dx² (2y(dy/dx)) = 2
Using the product rule, d²/dx² (2y(dy/dx)) = 2[(dy/dx) + y(d²y/dx²)] 2[(dy/dx) + y(d²y/dx²)] = 2 (dy/dx) + y(d²y/dx²) = 1
Therefore, the second derivative of the function y² = x² is (dy/dx) + y(d²y/dx²) = 1.
Explanation
We use the step-by-step process, starting with the first derivative. Using the product rule, we differentiate again to find the second derivative, then simplify the terms to find the final answer.
Problem 4
Prove: d/dx (y³) = 3y²(dy/dx).
Let’s start using the chain rule:
Consider y³ = (y)³
To differentiate, we use the chain rule: d/dx (y³) = 3y²(dy/dx)
The derivative of the outer function is multiplied by the derivative of the inner function.
Hence, d/dx (y³) = 3y²(dy/dx) is proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. As a final step, we show the substitution of the derivative of the inner function into the equation to derive the final result.
Problem 5
Solve: d/dx (y²/x)
To differentiate the function, we use the quotient rule: d/dx (y²/x) = (d/dx (y²) · x - y² · d/dx(x))/x²
We will substitute d/dx (y²) = 2y(dy/dx) and d/dx (x) = 1 = (2y(dy/dx) · x - y² · 1)/x² = (2yx(dy/dx) - y²)/x²
Therefore, d/dx (y²/x) = (2yx(dy/dx) - y²)/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 y²
1.Find the derivative of y².
2.Can we use the derivative of y² in real life?
3.Is it possible to take the derivative of y² at the point where y = 0?
4.What rule is used to differentiate y²/x?
5.Are the derivatives of y² and y³ the same?
Important Glossaries for the Derivative of y²
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Chain Rule: A rule for differentiating composite functions by differentiating the outer function and multiplying by the derivative of the inner function.
- Product Rule: A rule for differentiating products of two functions by adding the derivative of each function multiplied by the other function.
- Quotient Rule: A rule for differentiating quotients of two functions by applying a specific formula involving the derivatives of the numerator and denominator.
- Higher-Order Derivative: A derivative obtained by differentiating a function multiple times, indicating the rate of change of the rate of change.
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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.
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