Question

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

Answer 1

`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)`