Derivative of tan(xy)
2026-02-28 06:04 Diff
https://brightchamps.com/en-us/math/calculus/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: -

  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).