How do you use the chain rule to differentiate #y=-2csc^6x#?

1 Answer
Perfectly Good C.
Nov 21, 2017

#dy/dx=dy/(dg)(dg)/dx#, where #g(x)# is #cscx#

Explanation:

Make the substitution #g(x)=cscx#
#y=-2g^6(x)#

Now use the chain rule
#dy/dx=dy/(dg)(dg)/dx#
#dy/(dg)=-12g^5(x)#
#(dg)/dx=(cscx)'=-cscxcotx#
Plug these values in the initial equation
#dy/dx=dy/(dg)(dg)/dx=-12g^5(x)g'(x)#
#dy/dx=12csc^5xcscxcotx#
Therefore, the solution is
#dy/dx=12csc^7xcosx#

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