Derivative of Inverse Hyperbolic Tangent Function with Example

Understanding Inverse Hyperbolic Trig Derivatives

• 💡 The video focuses on finding the derivative of composite inverse hyperbolic trig functions.
• 🔑 Key formulas are provided: the derivative of inverse hyperbolic sine is u' / sqrt(u^2 + 1), and for inverse hyperbolic cosine, it's u' / sqrt(u^2 - 1).

Solving a Specific Derivative Problem

• 🎯 The problem involves finding the derivative of the inverse hyperbolic sine of tangent x.
• ⚡ Here, u is the inner function tan(x), and its derivative u' is sec^2(x).
• 🧩 Plugging these into the formula yields sec^2(x) / sqrt(tan^2(x) + 1).

Simplifying the Result Using Trig Identities

• 🛠️ The expression can be simplified using trigonometric identities, specifically 1 + tan^2(x) = sec^2(x).
• 📈 Substituting this identity into the denominator gives sqrt(sec^2(x)), which simplifies to sec(x).
• ✅ The final simplified derivative is sec(x).
• 📚 The importance of knowing fundamental trigonometric identities for simplification is emphasized.