How solve Derivatives of Trigonometric Functions?

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Answer 1 · 2018-06-18T13:50:51

#(df)/dx = 3 sqrt(csc(6x)) sin^2(4x) [4cos(4x) -cot(6x)sin(4x)] #

First use the product rule:

#(df)/dx = sqrt(csc(6x)) d/dx (sin^3(4x)) + d/dx (sqrt(csc(6x)) )sin^3(4x)#

using now the chain rule:

#(df)/dx = 3sqrt(csc(6x)) sin^2(4x) d/dx (sin(4x)) + d/dx (csc(6x)) (sin^3(4x))/(2sqrt(csc(6x)) ) #

#(df)/dx = 12sqrt(csc(6x)) sin^2(4x)cos(4x) -6cot(6x)csc(6x)(sin^3(4x))/(2sqrt(csc(6x)) ) #

and as the function is defined only for #cscx >=0 # we can simplify:

#(df)/dx = 12sqrt(csc(6x)) sin^2(4x)cos(4x) -3cot(6x)sqrt(csc(6x))sin^3(4x) #

#(df)/dx = 3 sqrt(csc(6x)) sin^2(4x) [4cos(4x) -cot(6x)sin(4x)] #