If #f(x)= tan5 x # and #g(x) = -x^2 -1 #, how do you differentiate #f(g(x)) # using the chain rule?

1 Answer
Tony B
Jul 9, 2016

#f^'(g(x))=-10x sec^2(5x)#

Explanation:

Known:# d/(dx)(tan x) = sec^2x# so # d/(dx)(tan 5x) = 5sec^2x#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let #g(x)=u=-x^2-1 -> (du)/(dx)=-2x#

Thus #f(g(x)) =y= tan(5u) -> (dy)/(du)=5sec^2(u)#
,.....................................................................................................

However:#" "(du)/(dx)xx(dy)/(du)" " =" " (du)/(du)xx(dy)/(dx)" "=" "(dy)/(dx)#

#=>(dy)/(dx)=-2x xx5sec^2(5x)#

#=>(dy)/(dx)=-10x sec^2(5x)#

Or if you prefer

#f^'(g(x))=-10x sec^2(5x)#

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