What is the derivative of #cos^-1(x)#?

1 Answer
Feb 7, 2017

Answer:

# d/dxcos^(-1)(x) = -1/sqrt(1 -x^2) #

Explanation:

When tackling the derivative of inverse trig functions. I prefer to rearrange and use Implicit differentiation as I always get the inverse derivatives muddled up, and this way I do not need to remember the inverse derivatives. If you can remember the inverse derivatives then you can use the chain rule.

Let #y=cos^(-1)(x) <=> cosy=x #

Differentiate Implicitly:

# -sinydy/dx = 1 # ..... [1]

Using the #sin"/"cos# identity;

# sin^2y+cos^2y -= 1 #
# :. sin^2y+x^2 = 1 #
# :. sin^2y = 1 -x^2#
# :. siny = sqrt(1 -x^2)#

Substituting into [1]

# :. -sqrt(1 -x^2)dy/dx=1 #
# :. dy/dx = -1/sqrt(1 -x^2) #