How do you differentiate given #f (x) = 3 arcsin (x^4)#?

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Answer 1 · 2016-11-13T19:38:54

# f'(x) = (12x^3)/sqrt(1-x^8) #

Let # y = f(x) = 3arcsin(x^4) , Then

# y/3 = arcsin(x^4) #
# sin(y/3) = x^4 # ..... [1]

We can now differentiate implicitly to get:

# cos(y/3)(1/3)dy/dx = 4x^3 #
# :. cos(y/3)dy/dx = 12x^3 # ..... [2]

Using the fundamental trig identity #sin^2A+cos^2A-=1# we can write:

#sin^2(y/3)+cos^2(y/3)=1#
#(x^4)^2+cos^2(y/3)=1# (from [1])
#cos^2(y/3)=1-x^8#
#cos(y/3)=sqrt(1-x^8)#

Substituting into [2] we get:

# sqrt(1-x^8)dy/dx = 12x^3 #

# dy/dx = (12x^3)/sqrt(1-x^8) #