What is the derivative of this function #sin^-1(x/4)#?

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Answer 1 · 2016-11-28T19:36:19

# d/dx sin^-1(x/4)= 1/(4sqrt(1-x^2/16)) #

Let # y=sin^-1(x/4) <=> siny=x/4 #

Differentiate Implicitly:

# cosydy/dx = 1/4 # .... [1]

Using the fundamental Trig Identity #sin^2A+cos^2A-=1#

# sin^2y+cos^2y=1#
# :. (x/4)^2+cos^2y=1#
# :. x^2/16+cos^2y=1#
# :. cos^2y=1-x^2/16#
# :. cosy=sqrt(1-x^2/16) #

Substituting into [1]:

# sqrt(1-x^2/16)dy/dx = 1/4 #
# :. dy/dx = 1/(4sqrt(1-x^2/16)) #