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105 LearnersLast updated on October 9, 2025
Derivative of 6xy
We use the derivative of 6xy to understand how this function changes in response to a slight change in x and y. Derivatives help us calculate various quantities such as rates of change in real-life situations. We will now discuss the derivative of 6xy in detail.
What is the Derivative of 6xy?
The derivative of 6xy with respect to x is found using the product rule, as 6xy consists of the product of two functions of x. The derivative is commonly represented as d/dx (6xy).
The key concepts are mentioned below:
Product Rule: The rule for differentiating a product of two functions.
Partial Derivative: Since this is a function of two variables, partial derivatives can be used to find the derivative with respect to one variable while holding the other constant.
Derivative of 6xy Formula
The derivative of 6xy with respect to x can be denoted as d/dx (6xy).
Using the product rule, the formula is: d/dx (6xy) = 6(y + x(dy/dx))
This formula applies to all x and y where the derivatives exist.
Proofs of the Derivative of 6xy
We can derive the derivative of 6xy using proofs. To show this, we will use differentiation rules.
One method to prove this is:
Using the Product Rule
The function 6xy is a product of two functions: 6x and y. We apply the product rule, which states: d/dx [u·v] = u'·v + u·v' Let u = 6x and v = y. Then: u' = d/dx (6x) = 6 v' = dy/dx Using the product rule: d/dx (6xy) = (6)(y) + (6x)(dy/dx) = 6y + 6x(dy/dx)
Higher-Order Derivatives of 6xy
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 (first derivative) and the rate of change of speed (second derivative) also change. Higher-order derivatives help us understand changes in functions like 6xy.
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 6xy, we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change.
Special Cases:
When x or y is zero, the derivative simplifies since terms involving x or y become zero. If y is a constant, the derivative becomes 6y.
Common Mistakes and How to Avoid Them in Derivatives of 6xy
Students frequently make mistakes when differentiating 6xy. 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 product rule correctly
Students may forget to apply the product rule correctly, leading to incorrect results. Ensure that each term is differentiated correctly and combined according to the product rule.
It's important to write each step in order to avoid mistakes.
Mistake 2
Confusing independent and dependent variables
Students might confuse x and y as independent variables when differentiating. Clarify which variable you are differentiating with respect to and treat other variables accordingly.
In partial derivatives, treat all other variables as constants.
Mistake 3
Incorrect simplification
Errors occur when simplifying the derivative. For example, failing to combine like terms or misapplying algebraic simplifications.
Double-check your work to ensure all terms are simplified correctly.
Mistake 4
Ignoring constant terms
There is a common mistake where students ignore constant coefficients like the 6 in 6xy. Ensure that constants are multiplied into the derivative correctly.
For example, do not ignore the 6 when applying the product rule.
Mistake 5
Neglecting to find partial derivatives
In multivariable calculus, students may overlook partial derivatives when necessary.
Remember that for functions of multiple variables, partial derivatives are essential in finding changes with respect to each variable.
Examples Using the Derivative of 6xy
Problem 1
Calculate the derivative of (6xy + x²).
Here, we have f(x, y) = 6xy + x². Using the product rule for 6xy and the power rule for x²: f'(x) = d/dx (6xy) + d/dx (x²) = 6(y + x(dy/dx)) + 2x So, the derivative of the specified function is 6y + 6x(dy/dx) + 2x.
Explanation
We find the derivative of the given function by applying the product rule to 6xy and the power rule to x².
Each term is differentiated separately, and the results are combined to get the final derivative.
Problem 2
A rectangular garden's area is represented by 6xy square meters, where x is the length and y is the width. If the length increases at a rate of 2 meters per second, find the rate of change of the area when the width is 5 meters.
We have the area A = 6xy. Taking the derivative with respect to time t, we use the product rule: dA/dt = 6(y(dx/dt) + x(dy/dt)) Given dx/dt = 2 m/s and y = 5 m, and assuming dy/dt = 0 (width constant): dA/dt = 6(5)(2) + 6x(0) = 60 m²/s Hence, the area increases at a rate of 60 square meters per second when the length increases.
Explanation
We use the product rule to differentiate the area function with respect to time.
By substituting the given rates and values, we find the rate of change of the area when the length changes.
Problem 3
Derive the second derivative of the function 6xy with respect to x.
The first step is to find the first derivative: d/dx (6xy) = 6(y + x(dy/dx))...(1) Now we differentiate equation (1) with respect to x to get the second derivative: d²/dx² (6xy) = 6(dy/dx + (dy/dx + x(d²y/dx²))) = 6(2(dy/dx) + x(d²y/dx²)) Therefore, the second derivative of the function 6xy with respect to x is 6(2(dy/dx) + x(d²y/dx²)).
Explanation
We start with the first derivative and differentiate it again with respect to x.
Using the product rule, we derive the second derivative by differentiating each component.
Problem 4
Prove: d/dx (6xy²) = 12xy(dy/dx) + 6y².
Let’s start using the product rule: Consider f(x, y) = 6xy² To differentiate, we use the product rule: d/dx (6xy²) = 6(d/dx (xy²)) = 6(y² + 2xy(dy/dx)) = 6y² + 12xy(dy/dx) Hence proved.
Explanation
We use the product rule to differentiate the function.
By isolating terms and applying the derivative rules, we derive the final result.
Problem 5
Solve: d/dx (6xy/x²).
To differentiate the function, we use the quotient rule: d/dx (6xy/x²) = (x²(d/dx (6xy)) - 6xy(d/dx (x²))) / x⁴ = (x²(6y + 6x(dy/dx)) - 6xy(2x)) / x⁴ = (6x²y + 6x³(dy/dx) - 12x²y) / x⁴ = (6x³(dy/dx) - 6x²y) / x⁴ = 6x(dy/dx)/x² - 6y/x² Therefore, d/dx (6xy/x²) = 6(dy/dx)/x - 6y/x²
Explanation
We use the quotient rule to differentiate the given function, applying the derivative rules for each term.
Finally, we simplify the expression to obtain the result.
FAQs on the Derivative of 6xy
1.Find the derivative of 6xy.
2.Can we use the derivative of 6xy in real life?
3.Is it possible to take the derivative of 6xy if y is a constant?
4.What rule is used to differentiate 6xy/x²?
5.Are the derivatives of 6xy and 6yx the same?
Important Glossaries for the Derivative of 6xy
- Derivative: The derivative of a function measures how the function changes as its input changes.
- Product Rule: A differentiation rule used to find the derivative of the product of two functions.
- Partial Derivative: The derivative of a function with respect to one variable while keeping other variables constant.
- Quotient Rule: A rule for differentiating the quotient of two functions.
- Rate of Change: A measure of how a quantity changes with respect to another quantity, often found using derivatives.
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Jaskaran Singh Saluja
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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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