How do you use chain rule to find the derivative of #y = 2 csc^3(sqrt(x)) #?

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Answer 1 · 2015-03-01T01:13:17

You can easily illustrate the chain rule using Leibniz notation

Let #y=2u^3# then #dy/(du)=6u^2 #

Let #u=csc(w) # then #(du)/(dw)=-csc(w)cot(w) #

Let #w=sqrt(x) # then #(dw)/(dx)=1/(2sqrt(x)) #

Now the chain rule is

#dy/dx=(dy)/(du)(du)/(dw)(dw)/(dx) #

#dy/dx=6u^2(-csc(w)cot(w))1/(2sqrt(x)) #

Remember that #u=csc(w) # and #w=sqrt(x)#

#dy/dx=6csc^2(sqrt(x))(-csc(sqrt(x))cot(sqrt(x)))(1/(2sqrt(x))) #

Some simplifying

#dy/dx=(-3csc^3(sqrt(x))cot(sqrt(x)))/sqrt(x) #