Derivative of inverse trigonometric functions.

To find the inverse functions,
if y = f(x) then x = f-1(y) or
if x = g(y) then y = g-1(x) or

use dy/dx = 1/(dx/dy)

Derivative of sin-1 x

y = sin-1 x
x = sin y
dy/dx = 1/(dx/dy)
dy/dx = 1/d(sin y)/dy = 1/cos y = 1/√(1 - x2)
d(sin-1 x)/dx = 1/√(1 - x2)

Derivative of cos-1 x

y = cos-1 x
x = cos y
dy/dx = 1/(dx/dy)
dy/dx = 1/d(cos y)/dy = -1/sin y = -1/√(1 - x2)
d(cos-1 x)/dx = -1/√(1 - x2)

Derivative of tan-1 x

y = tan-1 x
x = tan y
dy/dx = 1/(dx/dy)
dy/dx = 1/d(tan y)/dy = 1/sec2 y = cos2 y = 1/(1 + x2)
d(tan-1 x)/dx = 1/(1 + x2)


Derivative of cot-1 x

y = cot-1 x
x = cot y
dy/dx = 1/(dx/dy)
dy/dx = 1/d(cot y)/dy = -1/cosec2 y = -sin2 y = -1/(1 + x2)
d(cot-1 x)/dx = -1/(1 + x2)


Derivative of sec-1 x

y = sec-1 x
x = sec y
dy/dx = 1/(dx/dy)
dy/dx = 1/d(sec y)/dy = 1/sec y tan y = 1/x√(x2 - 1)
d(sec-1 x)/dx = 1/√(x2 - 1)


Derivative of cosec-1 x

y = cosec-1 x
x = cosec y
dy/dx = 1/(dx/dy)
dy/dx = 1/d(cosec y)/dy = -1/cosec y cot y = -1/x√(x2 - 1)
d(cosec-1 x)/dx = -1/√(x2 - 1)




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