Derivatives of Inverse Trigonometric Functions. Class 12 Math.

The derivatives of the common inverse trigonometric functions are: d/dx(sin⁻¹x) = 1/√(1-x²), d/dx(cos⁻¹x) = -1/√(1-x²), d/dx(tan⁻¹x) = 1/(1+x²), d/dx(cot⁻¹x) = -1/(1+x²), d/dx(sec⁻¹x) = 1/(|x|√(x²-1)), and d/dx(csc⁻¹x) = -1/(|x|√(x²-1)). These formulas are derived from implicit differentiation and the chain rule, applying the relationship that if y = f⁻¹(x), then x = f(y).
Here is a summary of the derivatives:
Inverse Sine:
Function: y = sin⁻¹x or y = arcsin(x)
Derivative: dy/dx = 1/√(1-x²)
Inverse Cosine:
Function: y = cos⁻¹x or y = arccos(x)
Derivative: dy/dx = -1/√(1-x²)
Inverse Tangent:
Function: y = tan⁻¹x or y = arctan(x)
Derivative: dy/dx = 1/(1+x²)
Inverse Cotangent:
Function: y = cot⁻¹x or y = arccot(x)
Derivative: dy/dx = -1/(1+x²)
Inverse Secant:
Function: y = sec⁻¹x or y = arcsec(x)
Derivative: dy/dx = 1/(|x|√(x²-1))
Inverse Cosecant:
Function: y = csc⁻¹x or y = arccsc(x)
Derivative: dy/dx = -1/(|x|√(x²-1))
These derivatives can be generalized for a composite function u = g(x). For example, the derivative of y = sin⁻¹(u) would be dy/dx = (1/√(1-u²)) * (du/dx).