How do you differentiate #f(x)=(x^4 - 2x^2)^6# using the chain rule?

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Answer 1 · 2015-10-30T19:54:35

#24x(x^4-2x^2)^5* (x^2-1)#

The chain rule is used to differentiate composite functions, and your two functions are

#{(f(x)=x^6),(g(x)=x^4-2x^2):}#

And your composition is #f(g(x))#.

The chain rule tells you that the derivative of #f(g(x))# is #f'(g(x)) * g'(x)#, and we have that

#{(f'(x)=6x^5),(g'(x)=4x^3-4x):}#

So:

#f'(g(x)) * g'(x) = 6(x^4-2x^2)^5 * (4x^3-4x)#.

Since we can factor #4x# from the last factor, we can rewrite it as

#24x(x^4-2x^2)^5* (x^2-1)#