105 LearnersLast updated on July 22nd, 2025
Derivative of tan(xy)
We use the derivative of tan(xy), which involves applying the product and chain rules, to understand how the tangent function changes in response to slight changes in both x and y. Derivatives help us calculate rates of change in various contexts. We will now discuss the derivative of tan(xy) in detail.
What is the Derivative of tan(xy)?
We now explore the derivative of tan(xy). It is represented as d/dx (tan(xy)) or (tan(xy))', and its calculation requires using the chain rule and other differentiation techniques. The function tan(xy) has a well-defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Tangent Function: tan(xy) = sin(xy)/cos(xy).
Chain Rule: Used for differentiating composite functions like tan(xy).
Product Rule: Applied when differentiating the product of x and y.
Derivative of tan(xy) Formula
The derivative of tan(xy) can be denoted as d/dx (tan(xy)) or (tan(xy))'. The formula we use to differentiate tan(xy) involves both the chain and product rules: d/dx (tan(xy)) = sec²(xy) * (y + x * dy/dx).
This formula applies to all x and y where cos(xy) ≠ 0.
Proofs of the Derivative of tan(xy)
We can derive the derivative of tan(xy) using proofs. To show this, we will use trigonometric identities along with the rules of differentiation. Different methods to prove this include: -
- By First Principle
- Using Chain Rule
- Using Product Rule
We will now demonstrate the differentiation of tan(xy) using the above methods:
By First Principle
The derivative of tan(xy) can be approached using the First Principle, expressing the derivative as the limit of the difference quotient. Consider f(x) = tan(xy). Its derivative can be expressed as: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h
Given f(x) = tan(xy), we substitute f(x + h) = tan((x + h)y).
Substituting these into the limit, we simplify using trigonometric identities and limits to obtain: f'(x) = sec²(xy) * (y + x * dy/dx).
Thus, the derivative is sec²(xy) * (y + x * dy/dx).
Using Chain Rule
To prove the differentiation of tan(xy) using the chain rule, we approach it as follows: Let u = xy, so tan(u) = sin(u)/cos(u).
Differentiating tan(u) with respect to u gives: d/dx (tan(xy)) = sec²(xy) * d/dx (xy).
Using the product rule on d/dx (xy), we get: d/dx (xy) = y + x * dy/dx.
Thus, d/dx (tan(xy)) = sec²(xy) * (y + x * dy/dx).
Using Product Rule
We use the formula tan(xy) = sin(xy)/cos(xy) and apply the product rule: Let u = sin(xy) and v = cos(xy).
Using the quotient rule: d/dx [u/v] = [v * d/dx(u) - u * d/dx(v)] / v².
Differentiating u and v using the chain and product rules, and substituting back,
we simplify to find: d/dx (tan(xy)) = sec²(xy) * (y + x * dy/dx).
Higher-Order Derivatives of tan(xy)
When a function is differentiated multiple times, the results are referred to as higher-order derivatives. Higher-order derivatives can be complex, but they provide deeper insights into the behavior of functions like tan(xy).
For the first derivative of a function, we write f′(x), indicating how the function changes at a specific point. The second derivative, f′′(x), is derived from the first derivative, indicating the rate of change of the rate of change.
Similarly, this pattern continues for higher orders, expressed as fⁿ(x).
Special Cases:
When xy = π/2, the derivative is undefined because tan(xy) has a vertical asymptote there. When xy = 0, the derivative of tan(xy) = sec²(0), which is 1.
Common Mistakes and How to Avoid Them in Derivatives of tan(xy)
Students frequently make mistakes when differentiating tan(xy). 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, leading to incorrect results. They often skip steps and directly arrive at the result, especially when solving using the product or chain rule. Ensure each step is written in order. Although it may seem tedious, it is essential to avoid errors.
Mistake 2
Forgetting the Undefined Points of tan(xy)
Students might not remember that tan(xy) is undefined at points like (xy = π/2, 3π/2,...). Consider the domain of the function you differentiate to understand that it is not continuous at these points.
Mistake 3
Incorrect use of Chain Rule
While differentiating functions like tan(xy), students might misapply the chain rule. For example, incorrect differentiation could occur when not accounting for both x and y changing. Always ensure the chain rule is applied correctly, considering both variables.
Mistake 4
Not writing Constants and Coefficients
A common mistake is forgetting to multiply the constants before tan(xy). For example, incorrectly writing d/dx (5tan(xy)) = sec²(xy) without the constant. Always check and multiply constants properly, such as d/dx (5tan(xy)) = 5sec²(xy) * (y + x * dy/dx).
Mistake 5
Not Applying the Product Rule
Students often forget to use the product rule when differentiating expressions involving products of variables, like xy. To avoid this, divide the expression into parts and apply the product rule correctly, ensuring each part is differentiated.
Examples Using the Derivative of tan(xy)
Problem 1
Calculate the derivative of tan(xy) * sec²(xy)
Consider f(x) = tan(xy) * sec²(xy).
Using the product rule, f'(x) = u′v + uv′ Here, u = tan(xy) and v = sec²(xy).
Differentiating each term: u′ = d/dx (tan(xy)) = sec²(xy) * (y + x * dy/dx) v′ = d/dx (sec²(xy)) = 2sec²(xy)tan(xy) * (y + x * dy/dx)
Substituting into the equation: f'(x) = [sec²(xy) * (y + x * dy/dx)] * sec²(xy) + tan(xy) * [2sec²(xy)tan(xy) * (y + x * dy/dx)]
Simplifying: f'(x) = sec⁴(xy) * (y + x * dy/dx) + 2tan²(xy)sec²(xy) * (y + x * dy/dx)
Thus, the derivative is sec⁴(xy) * (y + x * dy/dx) + 2tan²(xy)sec²(xy) * (y + x * dy/dx).
Explanation
We find the derivative by dividing the function into two parts. First, we differentiate each term and then combine them using the product rule to get the final result.
Problem 2
A company models its revenue using the function R = tan(xy), where x represents time in months and y represents the number of products sold. Calculate the rate of change of revenue when x = 2 months and y = 5.
Consider R = tan(xy) (revenue model)...(1)
Differentiating equation (1), dR/dx = sec²(xy) * (y + x * dy/dx)
Substitute x = 2 and y = 5: dR/dx = sec²(2*5) * (5 + 2 * dy/dx)
Simplifying, sec²(10) = 1/cos²(10), and assuming dy/dx is known from context or approximated, dR/dx = sec²(10) * (5 + 2 * dy/dx)
This expression represents the rate of change of revenue at x = 2 months and y = 5 products.
Explanation
We calculate the rate of change of revenue by differentiating the given function with respect to x and substituting the values of x and y. This helps determine how revenue changes with time and sales.
Problem 3
Derive the second derivative of the function R = tan(xy).
First, find the first derivative, dR/dx = sec²(xy) * (y + x * dy/dx)...(1)
Now differentiate equation (1) to find the second derivative: d²R/dx² = d/dx [sec²(xy) * (y + x * dy/dx)]
Using the product rule, d²R/dx² = [2sec²(xy)tan(xy) * (y + x * dy/dx) + sec²(xy) * d/dx(y + x * dy/dx)]
Simplifying further, d²R/dx² = 2sec²(xy)tan(xy) * (y + x * dy/dx) + sec²(xy) * (dy/dx + x * d²y/dx²)
Thus, the second derivative of R = tan(xy) is a combination of these terms.
Explanation
We start with the first derivative, then differentiate again using product and chain rules. This method provides the second derivative, revealing how the rate of change itself changes.
Problem 4
Prove: d/dx (tan²(xy)) = 2tan(xy)sec²(xy) * (y + x * dy/dx).
Using the chain rule: Let y = tan²(xy) = [tan(xy)]²
Differentiating, dy/dx = 2tan(xy) * d/dx [tan(xy)]
Since d/dx (tan(xy)) = sec²(xy) * (y + x * dy/dx), dy/dx = 2tan(xy) * sec²(xy) * (y + x * dy/dx)
Thus, d/dx (tan²(xy)) = 2tan(xy)sec²(xy) * (y + x * dy/dx)
Hence proved.
Explanation
We use the chain rule to differentiate the expression, replacing tan(xy) with its derivative. The step-by-step process leads to the final result.
Problem 5
Solve: d/dx (tan(xy)/x).
To differentiate, use the quotient rule: d/dx (tan(xy)/x) = (x * d/dx(tan(xy)) - tan(xy) * d/dx(x)) / x²
Substituting d/dx(tan(xy)) = sec²(xy) * (y + x * dy/dx) and d/dx(x) = 1, d/dx (tan(xy)/x) = (x * sec²(xy) * (y + x * dy/dx) - tan(xy)) / x²
Simplifying, = (x * sec²(xy) * (y + x * dy/dx) - tan(xy)) / x²
Thus, d/dx (tan(xy)/x) = (x * sec²(xy) * (y + x * dy/dx) - tan(xy)) / x².
Explanation
We differentiate the given function using the quotient rule and simplify to obtain the final result, accounting for the contributions of both the numerator and denominator.
FAQs on the Derivative of tan(xy)
1.Find the derivative of tan(xy).
2.Can we use the derivative of tan(xy) in real life?
3.Is it possible to take the derivative of tan(xy) at the point where xy = π/2?
4.What rule is used to differentiate tan(xy)/x?
5.Are the derivatives of tan(xy) and tan⁻¹(xy) the same?
6.Can we find the derivative of the tan(xy) formula?
Important Glossaries for the Derivative of tan(xy)
- Derivative: The derivative of a function indicates how the function changes in response to changes in its variables.
- Chain Rule: A technique used in calculus for differentiating composite functions.
- Product Rule: A rule used to differentiate the product of two functions.
- Quotient Rule: A method for differentiating the quotient of two functions.
- Tangent Function: A primary trigonometric function, expressed as tan(xy) in this context.
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.
