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.

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.

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: -

1. By First Principle 
2. Using Chain Rule 
3. 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).

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).

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.

• 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.