Derivatives of Inverse Trigonometric Functions

Here are the standard formulas for the derivatives of the three main inverse trigonometric functions:

• Derivative of arcsin(x) or sin^-1(x): d/dx (arcsin x) = 1 ÷ √(1 - x2) (for -1 < x < 1)
• Derivative of arccos(x) or cos^-1(x): d/dx (arccos x) = -1 ÷ √(1 - x2) (for -1 < x < 1)
• Derivative of arctan(x) or tan^-1(x): d/dx (arctan x) = 1 ÷ (1 + x2) (for all real x)

These derivatives are derived using the formula for the derivative of an inverse function or by using implicit differentiation and trigonometric identities.

**Example (Deriving arcsin x):**

1. Let y = arcsin(x).
2. This means sin(y) = x, where -π/2 ≤ y ≤ π/2.
3. Differentiate implicitly with respect to x: d/dx (sin y) = d/dx (x), which gives cos(y) * dy/dx = 1.
4. Solve for dy/dx: dy/dx = 1 ÷ cos(y).
5. Use the identity sin2y + cos2y = 1, so cos(y) = ±√(1 - sin2y). Since -π/2 ≤ y ≤ π/2, cos(y) is non-negative, so cos(y) = √(1 - sin2y).
6. Substitute sin(y) = x: cos(y) = √(1 - x2).
7. Substitute back into the derivative: dy/dx = 1 ÷ √(1 - x2).

You must apply the Chain Rule. If you have a function like h(x) = f(g(x)) where f(x) is an inverse trigonometric function and g(x) is the inner function, the Chain Rule gives:

h'(x) = f'(g(x)) * g'(x)

For example, to find the derivative of y = arcsin(x2):

• Outer function: arcsin(u), where u = x². Its derivative is 1 ÷ √(1 - u2).
• Inner function: g(x) = x². Its derivative is g'(x) = 2x.
• Apply Chain Rule: y' = [1 ÷ √(1 - (x2)2)] * 2x = 2x ÷ √(1 - x4).

Yes, you can look for patterns:

• **Pairs:** The derivatives of the "co" functions (arccos, arccot, arccsc) are the negative of the derivatives of their non-"co" counterparts (arcsin, arctan, arcsec).
• **Denominators:** Arcsin and arccos have the same denominator √(1 - x2). Arctan and arccot have the same denominator (1 + x2). Arcsec and arccsc also share a similar denominator pattern.
• **Structure:** Arcsin/arccos have square roots. Arctan/arccot do not.

Focus on memorizing one from each pair (arcsin, arctan, arcsec) and remember the "co" rule for the others.