What is the derivative of cos-1 (x)?
We’ll solve it using trigonometric identity sin2y + cos2y = 1.
Answer: The derivative of cos-1x is -1/√(1−x2)
Let see, how we can solve using this trigonometric identity.
We have to find the derivative of cos-1x.
Let us assume y = cos-1x. Then, cos y = x
Differentiate implicitly with respect to x.
(-sin y) dy/dx = 1 ————- (i)
By trigonometric identity, we know that
sin2y + cos2y = 1
⇒ sin2y + x2 = 1
⇒ sin2y = 1 – x2
⇒ sin y = √(1 − x2)
Substituting the above value in (i), we get
= −√(1 − x2) dy/dx = 1
⇒dy/dx = –1/√(1 − x2)