How do you differentiate #cos(x^4)#?

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Answer 1 · 2016-09-05T12:33:50

#- 4 x^(7)#

We have: # cos(x^(4))#

This expression can be differentiated using the "chain rule".

Let #u = x^(4) => u' = 4 x^(3)# and #v = cos(u) => v' = - sin(u)#:

#=> (d) / (dx) (cos(x^(4))) = 4 x^(3) cdot - sin(u)#

#=> (d) / (dx) (cos(x^(4))) = - 4 u x^(3)#

We can now replace #u# with #x^(4)#:

#=> (d) / (dx) (cos(x^(4))) = - 4 (x^(4)) x^(3)#

#=> (d) / (dx) (cos(x^(4))) = - 4 x^(4 + 3)#

#=> (d) / (dx) (cos(x^(4))) = - 4 x^(7)#