How do you differentiate #f(x)=(tan^2 3x)^(7/3)# using the chain rule?

1 Answer
Jim G.
Apr 5, 2016

# 42/3(tan^2 3x)^(4/3) tan3x sec^2 3x #

Explanation:

differentiate using the #color(blue)" chain rule " #

#d/dx [f(g(x)) ] = f'(g(x)) . g'(x) #
#"------------------------------------------------------------------------"#

f(g(x)) =#(tan^2 3x)^(7/3)rArr f'(g(x)) = 7/3(tan^2 3x)^(4/3) #

g(x) = #tan^2 3x rArr g'(x) = 2tan3x . d/dx(tan3x) #

#d/dx(tan3x) = sec^2 3x .d/dx(3x) = 3sec^2 3x #
#"-----------------------------------------------------------------------"#

Now combining these results together

f'(x) = #7/3(tan^2 3x)^(4/3).2tan3x.3sec^2 3x #

# = 42/3(tan^2 3x)^(4/3) tan3x sec^2 3x #

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