What is the derivative of this function #y=tan^-1(2x^4)#?

1 Answer
GreyWolf-X
May 30, 2017

#(8x^3)/(4x^8+1)#

Explanation:

Aside from just applying the chain rule, one helpful way to perform this derivation may be to substitute a variable u to break the equation into separate pieces of a puzzle, like this:

We know the chain rule is
#(df(u))/dx=(df)/(dx)*(du)/(dx)#

We can also let u (in this case) = #2x^4#

Now just take the derivative of the function in two puzzle pieces:

We know #d/(du)# #tan^-1(u)# is just #d/(dx)(Tan^-1 (x)) = 1/(x^2+1)#

SO...

#d/(du)(tan^-1(u)) = 1/(u^2+1)#

and

#d/(dx) (2x^4) = 8x^3# (By the power rule)

From here, just substitute back #u=2x^4# to get #1/((2x^4)^2+1)*8x^3#

Which simplifies to... #(8x^3)/(4x^8+1)#

V'oila! Our puzzle is complete!

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