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113 LearnersLast updated on September 27, 2025
Derivative of y² with Respect to x
The derivative of y² with respect to x is a fundamental concept in calculus, used to measure how the square of a variable y changes with respect to a change in x. Derivatives are crucial in calculating rates of change and finding slopes of curves in various real-life applications. We will now explore the derivative of y² in detail.
What is the Derivative of y² with Respect to x?
The derivative of y² with respect to x is represented as d/dx (y²) or (y²)'. To find this derivative, we use the chain rule, assuming y is a function of x.
The key concepts related to this derivative are:
Chain Rule: A rule for differentiating compositions of functions.
Power Rule: A rule for differentiating expressions of the form yⁿ.
Derivative of y² Formula
The derivative of y² with respect to x is found using the chain rule and the power rule. The formula is: d/dx (y²) = 2y(dy/dx)
This formula applies whenever y is a differentiable function of x.
Proofs of the Derivative of y²
We can derive the derivative of y² using proofs and fundamental rules of differentiation. Here are a few methods:
Using the Chain Rule
Considering y as a function of x, we have y² = (y(x))². Applying the chain rule, we differentiate: d/dx (y²) = d/dx [y(x)]² = 2y(x) * d/dx [y(x)] = 2y * dy/dx
Using the Power Rule
The power rule states that d/dx [uⁿ] = n * uⁿ⁻¹ * (du/dx) for any function u(x). Here, u(x) = y(x) and n = 2, so: d/dx (y²) = 2y * dy/dx Thus, in both methods, the derivative of y² with respect to x is confirmed as 2y(dy/dx).
Higher-Order Derivatives of y²
Higher-order derivatives involve differentiating a function multiple times. For example, the second derivative is the derivative of the first derivative.
For y², the second derivative can be found by differentiating 2y(dy/dx) again with respect to x. Higher-order derivatives help analyze more complex behavior, similar to understanding acceleration as the rate of change of velocity.
Special Cases
When y is a constant, the derivative of y² with respect to x is zero, since the derivative of a constant is zero.
If y is a linear function of x, the derivative of y² will be proportional to the derivative of y.
Common Mistakes and How to Avoid Them in Derivatives of y²
Students often make mistakes when finding the derivative of y² with respect to x. Understanding the correct steps can help avoid these errors. Here are some common mistakes and tips to solve them:
Mistake 1
Not Using the Chain Rule Properly
Students might forget to apply the chain rule when y is a function of x, leading to incomplete results.
Always remember to multiply by dy/dx when differentiating y².
Mistake 2
Confusing with Constant Functions
When y is a constant, students might assume the derivative of y² is non-zero.
Remember, if y is constant, y² is also constant, and its derivative is zero.
Mistake 3
Overlooking Higher-Order Derivatives
While solving for higher-order derivatives, students might overlook the need to apply the chain rule repeatedly.
Ensure each derivative is calculated with respect to x, accounting for all dependencies.
Mistake 4
Ignoring y as a Function of x
Sometimes, students treat y as independent of x when it should be considered a function of x.
Always verify if y depends on x before differentiating.
Mistake 5
Misapplying the Power Rule
The power rule should be applied correctly, treating y as a function of x.
Avoid errors by ensuring all steps and rules are followed correctly.
Examples Using the Derivative of y²
Problem 1
Calculate the derivative of (y²z) with respect to x.
Here, we have f(x) = y²z, where y and z are functions of x. Using the product rule, f'(x) = (d/dx (y²))z + y²(d/dx (z)) = (2y(dy/dx))z + y²(dz/dx) Thus, the derivative of the specified function is 2y(dy/dx)z + y²(dz/dx).
Explanation
We find the derivative by dividing the function into parts and applying the product rule.
First, differentiate each part, then combine them to get the final result.
Problem 2
A balloon's radius changes over time and is given by r(t) = t². Find the rate of change of the volume V = (4/3)πr³ with respect to time t when t = 1 second.
The volume V is given by V = (4/3)πr³. First, find dr/dt: r(t) = t², so dr/dt = 2t. Now, use the chain rule: dV/dt = dV/dr * dr/dt dV/dr = 4πr² So, dV/dt = 4πr² * 2t When t = 1, r = 1² = 1. Thus, dV/dt = 4π(1)² * 2(1) = 8π. The rate of change of volume at t = 1 second is 8π cubic units per second.
Explanation
We start by finding dr/dt, then use the chain rule to find dV/dt.
Finally, substituting the given time into the expression gives us the rate of change.
Problem 3
Derive the second derivative of the function y² with respect to x.
First, find the first derivative: d/dx (y²) = 2y(dy/dx) Now, differentiate again to get the second derivative: d²/dx² (y²) = d/dx [2y(dy/dx)] = 2[(dy/dx)² + y(d²y/dx²)] Therefore, the second derivative of y² is 2[(dy/dx)² + y(d²y/dx²)].
Explanation
We find the second derivative by differentiating the first derivative using the product rule, considering y as a function of x.
Problem 4
Prove: d/dx (y³) = 3y²(dy/dx).
Using the chain rule: Consider y³ = (y(x))³. d/dx (y³) = 3(y(x))² * d/dx [y(x)] = 3y² * dy/dx Hence proved.
Explanation
In this step-by-step process, we use the chain rule to differentiate y³, treating y as a function of x.
Problem 5
Solve: d/dx (y²/x).
To differentiate the function, use the quotient rule: d/dx (y²/x) = [(x)(d/dx (y²)) - (y²)(d/dx(x))]/x² = [x(2y(dy/dx)) - y²]/x² = (2xy(dy/dx) - y²)/x² Therefore, d/dx (y²/x) = (2xy(dy/dx) - y²)/x².
Explanation
In this process, we differentiate the given function using the quotient rule.
We simplify the equation to obtain the final result.
FAQs on the Derivative of y²
1.Find the derivative of y² with respect to x.
2.Can the derivative of y² be used in real life?
3.What happens if y is a constant?
4.What rule is used to differentiate y²/x?
5.Is the derivative of y³ the same as y²?
Important Glossaries for the Derivative of y²
- Derivative: The derivative measures how a function changes as its input changes.
- Chain Rule: A rule for differentiating compositions of functions, crucial when y is a function of x.
- Power Rule: A rule used to differentiate functions of the form yⁿ.
- Quotient Rule: A method for finding the derivative of a quotient of two functions.
- Higher-Order Derivative: Derivatives of derivatives, providing deeper insights into the behavior of a function.
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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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: He loves to play the quiz with kids through algebra to make kids love it.
