What is the derivative of #tan^4(3x)#?

1 Answer
Jim H
Apr 9, 2015

If is helpful (until you really get used to it) to remember and rewrite:
#tan^4(3x) = (tan(3x))^4#

This makes it clear that ultimately (at the last), this is a fourth power function. We'll need the power rule and the chain rule. (Some authors call this combination "The General Power Rule".) In fact we'll need the chain rule twice.

#d/(dx)((tan(3x))^4) = 4 (tan(3x))^3 [d/(dx)(tan(3x))]#

#=4tan^3(3x)[sec^2(3x) d/(dx)(3x)]=4tan^3(3x)[3sec^2(3x)]#

#d/(dx)(tan^4(3x))=12tan^3(3x) sec^2(3x)#

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