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

Answer 1

Elijah Abram

# 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)) #

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Answer 2

Scarlett Allison

The derivative of the function ( \sin^{-1}(x/4) ) is ( \frac{1}{\sqrt{16-x^2}} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.